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Hindawi Publishing CorporationJournal of Discrete MathematicsVolume 2013 Article ID 450481 10 pageshttpdxdoiorg1011552013450481
Research Article119902-Pascal and 119902-Wronskian Matrices with Implications to119902-Appell Polynomials
Thomas Ernst
Department of Mathematics Uppsala University PO Box 480 751 06 Uppsala Sweden
Correspondence should be addressed toThomas Ernst thomasmathuuse
Received 19 June 2012 Accepted 2 December 2012
Academic Editor Franck Petit
Copyright copy 2013 Thomas ErnstThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We introduce a 119902-deformation of the Yang andYounmatrix approach forAppell polynomialsThiswill lead to a powerfulmachineryfor producing new and old formulas for 119902-Appell polynomials and in particular for 119902-Bernoulli and 119902-Euler polynomialsFurthermore the 119902-119871-polynomial anticipated by Ward can be expressed as a sum of products of 119902-Bernoulli and 119902-Eulerpolynomials The pseudo 119902-Appell polynomials which are first presented in this paper enable multiple 119902-analogues of the Yangand Youn formulas The generalized 119902-Pascal functional matrix the 119902-Wronskian vector of a function and the vector of 119902-Appellpolynomials together with the 119902-deformed 120598 matrix multiplication from the authors recent article are the main ingredients in theprocess Beyond these results we give a characterization of 119902-Appell numbers improving on Al-Salam 1967 Finally we find a119902-difference equation for the 119902-Appell polynomial of degree 119899
1 Introduction
In this paper we will introduce the 119902-Pascal and 119902-Wronskianmatrices in a general setting with the aim of fruitful applica-tions for 119902-Appell polynomials These two matrices containa certain 119902-difference operator just like the 119902-deformedLeibniz functional matrix from [1] By the 119902-Leibniz formulaproducts of suchmatrices contain a certain operator sdot
120598 just as
in the previous article but with a slightly different definitionHowever since in almost all equations we compute thefunction value at 119905 = 0 this operator will convert to ordinarymatrixmultiplication by formula (6)There is a connection tothe 119902-Pascal matrix [2] for the special case 119891 = 119902-exponentialfunctionThe Appell polynomials are seldom encountered inthe literature one exception was Carlitzrsquo article [3] wherea formula for the product of two generating functions wasgiven Carlitz cleverly observed that the generating functionfor Appell polynomials can be inverted when 119891(0) = 1 TheNWA 119902-addition seldom occurs in the literature except forthe authorrsquos papers it was first seen in an Italian paper byNalli[4] Corresponding to the two dual 119902-additions NWA andJHC there are two dual 119902-Bernoulli polynomials with thesame names these are special cases of 119902-Appell polynomialsthe content of Section 2 It turns out that for certain purposes
wewill need pseudo 119902-Appell polynomials which correspondto the other basic 119902-addition JHC these polynomials werefirst seen in [5] The 119902-Appell numbers form a certainalgebraic structure with two operations which was firstdescribed by Al-Salam [6] This structure will in a certainway be generalized to 119902-Appell polynomials The 119902-Bernoulliand 119902-Euler polynomials are presented along with the 119902-Lucas polynomials these definitions coincide with thosegiven in [7] In Section 3 we come to the core of the paperwhich starts with the 119902-Pascal matrix and the 119902-Wronskianvector In the beginning we briefly pass to the shorter paper[8] to find a few 119902-analogues of truncated 119902-exponentialgenerating function formulas We mostly follow the orderingin [9] Sometimes multiple 119902-analogues follow and thesewill usually come with the pseudo 119902-Appell polynomialsA couple of general formulas for 119902-Appell polynomials aregiven sometimes these are specialized to 119902-Bernoulli and 119902-Euler polynomials In the last part we find a 119902-differenceequation for general 119902-Appell polynomials In an appendixwecompare our formulas for Bernoulli and Euler polynomialswith Hansenrsquos table [10] where another notation is used
We now start with the definition of the umbral methodfrom the recent book [11] Some of the notations are wellknown and will be skipped
2 Journal of Discrete Mathematics
Definition 1 Let the Gauss 119902-binomial coefficient be definedby
(119899
119896)
119902
equiv119899119902
119896119902119899 minus 119896
119902 119896 = 0 1 119899 (1)
Let 119886 and 119887 be any elements with commutativemultiplicationThen the NWA 119902-addition is given by
(119886 oplus119902119887)119899
equiv
119899
sum
119896=0
(119899
119896)
119902
119886119896
119887119899minus119896
119899 = 0 1 2 (2)
The JHC 119902-addition is the function
(119886 ⊞119902119887)119899
equiv
119899
sum
119896=0
(119899
119896)
119902
119902(119896
2)
119887119896
119886119899minus119896
= 119886119899
(minus119887
119886 119902)
119899
119899 = 0 1 2
(3)
The 119902-exponential function is defined by
E119902(119911) equiv
infin
sum
119896=0
1
119896119902119911119896
(4)
The 119902-derivative is defined by
(D119902119891) (119909) equiv
119891 (119909) minus 119891 (119902119909)
(1 minus 119902) 119909 if 119902 isin C 1 119909 = 0
119889119891
119889119909(119909) if 119902 = 1
119889119891
119889119909(0) if 119909 = 0
(5)
Assume that 119905119910 and 119902 are the variables If wewant to indicatethe variable on which the 119902-difference operator is applied wedenote the operator in (5) by (D
119902119905120593)(119905 119910) To save space we
sometimes write 1198911015840 instead of D119902119905119891
Theorem2 The following equation relates the 119899th 119902-differenceoperator at zero to the 119899th derivative at zero for an analyticfunction 119891(119911) [12]
D119899119902119891 (0) =
119899119902
119899119891(119899)
(0) (6)
For convenience we will use the following two abbrevia-tions for functions 119891(119905) 119891(119896) represents the 119896th 119902-derivativeof 119891 and 119891
119896 represents the 119896th power of 119891 We will onlyconsider functions in C[[119905]] Sometimes functions can begiven in other forms we then mean the Taylor expansion ofthe function around zero In the sense ofMilne-Thomson [13]and Rainville [14] we write = to indicate the symbolic nature
If 119891(119909) isin C[119909] the ring of polynomials with complexcoefficients then the function 120598 C[119909] 997891rarr C[119909] is definedby
120598119891 (119909) equiv 119891 (119902119909) (7)
We make the convention that all 119899 times 119899 matrices are denotedby the first index 119899 The rows and columns are denoted by 119894and 119895 The 119902-Pascal matrix [2] is defined by
119875119899119902(119909) (119894 119895) equiv (
119894
119895)
119902
119909119894minus119895
(8)
The Polya-Vein matrix119867119899119902
[2] is given by
119867119899119902(119894 119894 minus 1) equiv 119894
119902 119894 = 1 119899 minus 1
119867119899119902(119894 119895) equiv 0 119895 = 119894 minus 1
(9)
2 119902-Appell Polynomials
The Appell polynomials have been around for more than acentury but this general notion is still not so well knownTheusual definition is
119891 (119905) 119890119909119905
=
infin
sum
120584=0
119905120584
120584Φ120584(119909) 119891 (119905) isin C [[119909]] (10)
The original article by Appell [15] started with the definition119889119860119899119889119909 = 119899119860
119899minus1 Then Appell showed that multiplication of
two Appell polynomials is commutative He then consideredthe inverse of119860
119899as a natural generalization of ordinary divi-
sionThe 119902-Appell polynomials form a natural generalizationof the Appell polynomials
21 Definitions Weconsider a certain value of the order (= 1)for the 119902-Appell polynomials in [11] The degree is only aname it is not the degree of polynomials
Definition 3 The 119902-Appell polynomials Φ120584119902(119909) of degree 120584
are defined by
119891 (119905)E119902(119909119905) =
infin
sum
120584=0
119905120584
120584119902Φ120584119902(119909) 119891 (119905) isin C [[119909]] (11)
We say that Φ120584119902(119909) is the 119902-Appell polynomial sequence for
119891(119905)We put 119909 = 0 and obtain
119891 (119905) =
infin
sum
120584=0
119905120584
120584119902Φ120584119902 Φ
0119902= 1 (12)
where Φ120584119902
is said to have degree 120584
Definition 4 The pseudo 119902-Appell polynomials Φ119902
120584(119909) of
degree 120584 [5] are defined by
119891 (119905)E1119902
(119909119905) =
infin
sum
120584=0
119905120584
120584119902Φ119902
120584(119909) (13)
We say that Φ119902120584(119909) is the pseudo 119902-Appell polynomial seq-
uence for 119891(119905)By putting 119909 = 0 we have
119891 (119905) =
infin
sum
120584=0
119905120584
120584119902Φ119902
120584 (14)
where Φ119902120584is called a Φ119902 number of degree 120584
Journal of Discrete Mathematics 3
We have similarly
D119902Φ119902
120584(119909) = 120584
119902Φ119902
120584minus1119902(119902119909)
Φ119902
120584(119909) = (Φ
119902⊞119902119909)120584
(15)
22 119902-Analogue of Carlitzrsquo Product Formula (1963) To moti-vate the generalized 119902-Pascal functional matrix we show thatthe Carlitz [3 page 247] formula for the product of generatingfunctions for Appell polynomials has a 119902-analogue involvingthe NWA 119902-addition Define the numbers Φ1015840
120584119902infin
120584=0by the
inverse of (12) (this is allowable since 119891(0) = 1)
119891(119905)minus1
=
infin
sum
120584=0
119905120584
120584119902Φ1015840
120584119902 Φ1015840
0119902= 1 (16)
Then we have the following
Theorem 5 An inverse relation formula for the 119902-Appellpolynomials
119909119899
=
119899
sum
119896=0
(119899
119896)
119902
Φ1015840
119899minus119896119902Φ119896119902(119909) (17)
Proof By the umbral definition of 119902-Appell polynomials wehave
119899
sum
119896=0
(119899
119896)
119902
Φ1015840
119899minus119896119902Φ119896119902(119909)
=
119899
sum
119896=0
(119899
119896)
119902
Φ1015840
119899minus119896119902
119896
sum
119897=0
(119896
119897)
119902
Φ119896minus119897119902
119909119897
=
119899
sum
119896=0
119899119902Φ1015840
119899minus119896119902
119899 minus 119896119902
119899
sum
119897=0
Φ119896minus119897119902
119897119902119896 minus 119897
119902119909119897
=
119899
sum
119897=0
119899119902
119897119902119909119897
sum
119904+119905=119899minus119897
Φ1015840
119904119902Φ119905119902
119904119902119905119902
=
119899
sum
119897=0
119899119902
119897119902119909119897
120575119899minus119897
= 119909119899
(18)
This completes the proof
We introduce two other 119902-Appell polynomials Ξ119902and Ψ
119902
by
119892 (119905)E119902(119909119905) =
infin
sum
120584=0
119905120584
120584119902Ξ120584119902(119909) 119892 (119905) isin C [[119909]]
119892 (119905) =
infin
sum
120584=0
119905120584
120584119902Ξ120584119902 Ξ0119902
= 1
ℎ (119905)E119902(119909119905) =
infin
sum
120584=0
119905120584
120584119902Ψ120584119902(119909) ℎ (119905) isin C [[119909]]
ℎ (119905) =
infin
sum
120584=0
119905120584
120584119902Ψ120584119902 Ψ0119902
= 1
(19)
Let us consider119891119898(119886119909) 119892
119899(119887119909)
=
119898
sum
119903=0
119899
sum
119904=0
(119898
119903)
119902
(119899
119904)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
119909119903+119904
(20)
Now put
ℎ(119905)minus1
=
infin
sum
120584=0
119905120584
120584119902Ψ1015840
120584119902 (21)
Because of (17) we get
119891119898(119886119909) 119892
119899(119887119909)
=
119898
sum
119903=0
119899
sum
119904=0
(119898
119903)
119902
(119899
119904)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
119903+119904
sum
119895=0
(119903 + 119904
119895)
119902
Ψ1015840
119903+119904minus119895119902Ψ119895119902(119909)
(22)
Introduce the notation119860 (119895 119902119898 119899 119886 119887)
equiv
119898
sum
119903=0
119899
sum
119904=0
(119898
119903)
119902
(119899
119904)
119902
(119903 + 119904
119895)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
Ψ1015840
119903+119904minus119895119902
(23)
where
119891119898(119886119909) 119892
119899(119887119909) =
119898+119899
sum
119895=0
119860 (119895 119902119898 119899 119886 119887)Ψ119895119902(119909) (24)
Theorem 6 With the previously mentioned notation one hasthe following 119902-analogue of Carlitz [3 page 247]
infin
sum
119898=0
infin
sum
119899=0
119860 (119895 119902119898 119899 119886 119887)119906119898
119907119899
119898119902119899119902
= 119891 (119906) 119892 (119907)
infin
sum
119896=0
Ψ1015840
119896119902
(119886119906 oplus119902119887119907)119896+119895
119895119902119896119902
(25)
Proof Thus we haveinfin
sum
119898=0
infin
sum
119899=0
119860 (119895 119902119898 119899 119886 119887)119906119898
119907119899
119898119902119899119902
=
infin
sum
119898=0
infin
sum
119899=0
119906119898
119907119899
119898119902119899119902
119898
sum
119903=0
119899
sum
119904=0
= (119898
119903)
119902
(119899
119904)
119902
(119903 + 119904
119895)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
Ψ1015840
119903+119904minus119895119902
=
infin
sum
119903=0
infin
sum
119904=0
(119886119906)119903
(119887119907)119904
119903119902119904119902
(119903 + 119904
119895)
119902
Ψ1015840
119903+119904minus119895119902
infin
sum
119898=0
infin
sum
119899=0
119891119898119892119899
119906119898
119907119899
119898119902119899119902
= 119891 (119906) 119892 (119907)
infin
sum
119903=0
infin
sum
119904=0
(119886119906)119903
(119887119907)119904
119903119902119904119902
4 Journal of Discrete Mathematics
(119903 + 119904
119895)
119902
Ψ1015840
119903+119904minus119895119902
= 119891 (119906) 119892 (119907)
infin
sum
119896=119895
(119896
119895)
119902
Ψ1015840
119896minus119895119902sum
119903+119904=119896
(119886119906)119903
(119887119907)119904
119903119902119904119902
= 119891 (119906) 119892 (119907)
infin
sum
119896=119895
(119896
119895)
119902
Ψ1015840
119896minus119895119902
(119886119906 oplus119902119887119907)119896
119896119902
(26)
23 Characterization of 119902-Appell Numbers We will now pre-sent a characterization of 119902-Appell numbers comparewithAl-Salam [6]
Definition 7 We denote by A119902the set of all 119902-Appell-
numbersLet Φ
119899119902and Ψ
119899119902be two elements in A
119902 Then the
operations + and ⋆ are defined as follows
(Φ119902+ Ψ119902)119899
equiv Φ119899119902+ Ψ119899119902 (27)
(Φ119902⋆ Ψ119902)119899
equiv
119899
sum
119896=0
(119899
119896)
119902
Φ119899minus119896119902
Ψ119896119902 (28)
Theorem 8 The identity element in A119902with respect to ⋆ is
119868 equiv 1205751198990 This corresponds to 119891(119905) = 1
Proof Put 119896 = 119899 and Φ119902= 1 in (28)
Theorem 9 (Compare with [6 page 34]) Let Φ119899119902 Ξ119899119902
andΨ119899119902
be three elements in A119902corresponding to the generating
functions119891(119905) 119892(119905) and ℎ(119905) respectively Further assume that119891(0) + ℎ(0) = 0 Then
Φ119899119902+ Ψ119899119902
isin A119902
Φ119899119902+ Ψ119899119902
has generating function119891 (119905) + ℎ (119905) (29)
The associative law for addition reads
Φ119899119902+ (Ξ119899119902+ Ψ119899119902) = (Φ
119899119902+ Ξ119899119902) + Ψ119899119902 (30)
Proof This follows from the definitions
Theorem 10 (Compare with [6 page 34]) LetΦ119899119902 Ξ119899119902 and
Ψ119899119902
be three elements in A119902corresponding to the generating
functions 119891(119905) 119892(119905) and ℎ(119905) respectively Then
Φ119899119902⋆ Ψ119899119902
isin A119902 (31)
Φ119899119902⋆ Ψ119899119902
= Ψ119899119902⋆ Φ119899119902 (32)
Φ119899119902⋆ Ψ119899119902
has generating function119891 (119905) ℎ (119905) (33)
The associative law for ⋆ reads
Φ119899119902⋆ (Ξ119899119902⋆ Ψ119899119902) = (Φ
119899119902⋆ Ξ119899119902) ⋆ Ψ119899119902 (34)
Proof We prove (31) and (33) at the same time We have inthe umbral sense
119891 (119905) ℎ (119905) = E119902(119905 (Φ119902oplus119902Ψ119902)) (35)
Formulas (32) and (34) follow by the commutativity andthe associativity of NWA
24 Specializations
Definition 11 The NWA 119902-Bernoulli polynomialsBNWA120584119902(119909) of degree 120584 are defined by
1199051
(E119902(119905) minus 1)
E119902(119909119905) =
infin
sum
120584=0
119905120584BNWA120584119902 (119909)
120584119902
|119905| lt 2120587 (36)
The NWA 119902-Bernoulli polynomials are also given by
BNWA120584119902 (119909) = (BNWA119902 oplus119902 119909)120584
(37)
The pseudo-119902-Bernoulli polynomials are defined by
B119902NWA120584 (119909) = (BNWA119902 ⊞119902 119909)120584
(38)
The JHC 119902-Bernoulli polynomials BJHC120584119902(119909) of degree 120584are defined by
1199051
(E1119902
(119905) minus 1)E119902(119909119905) =
infin
sum
120584=0
119905120584BJHC120584119902 (119909)
120584119902
|119905| lt 2120587 (39)
3 119902-Pascal and 119902-Wronskian Matrices
We now come to the most important part of the paper whichis of interest even for the case 119902 = 1 since the original paperby Yang and Youn is not very widespread
Definition 12 The generalized 119902-Pascal functional matrix isgiven by
(P119899119902) [119891 (119905 119902)] (119894 119895)
equiv
(119894
119895)
119902
D119894minus119895119902119905119891 (119905 119902) if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(40)
In the special case 119891(119905 119902) = constant
(P119899119902) [119891 (119905 119902)] = 119891 (119905 119902) 119868 (41)
Definition 13 The 119902-deformed Wronskian vector is given by
(W119899119902) [119891 (119905 119902)] (119894 0) equiv D119894
119902119905119891 (119905 119902) 119894 = 0 1 119899 minus 1
(42)
Journal of Discrete Mathematics 5
Definition 14 The 119902-deformed Leibniz functional matrix isgiven by(L119899119902) [119891 (119905 119902)] (119894 119895)
equiv
D119894minus119895119902119905119891 (119905 119902)
119894 minus 119895119902
if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(43)
In order to find a 119902-analogue of the formula [16]
(L119899) [119891 (119905) 119892 (119905)] = (L
119899) [119891 (119905)] (L
119899) [119892 (119905)] (44)
we infer that by the 119902-Leibniz formula [1](L119899119902) [119891 (119905 119902) 119892 (119905 119902)]
= (L119899119902) [119891 (119905 119902)] sdot
120598(L119899119902) [119892 (119905 119902)]
(45)
where in the matrix multiplication for every term whichincludes D119896
119902119891 we operate with 120598119896 on 119892 We denote this by sdot
120598
This operator can also be iterated compare with [1]
Theorem 15 (a 119902-analogue of [9 page 67]) Consider
P119899119902[E119902(119909119905)]
10038161003816100381610038161003816119905=0= 119875119899119902(119909) (46)
where the 119902-Pascal matrix is defined in (8)The mappingsP
119899119902[sdot] andW
119899119902[sdot] are linear that is
P119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886P
119899119902[119891 (119905)] + 119887P
119899119902[119892 (119905)]
W119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886W
119899119902[119891 (119905)] + 119887W
119899119902[119892 (119905)]
(47)
One has the following two matrix multiplication formulasthe first one is a 119902-analogue of [8 page 232]
P119899119902[119891 (119905 119902)] sdot
120598P119899119902[119892 (119905 119902)] = P
119899119902[119891 (119905 119902) 119892 (119905 119902)]
(48)
P119899119902[119891 (119905 119902)] sdot
120598W119899119902[119892 (119905 119902)] = W
119899119902[119891 (119905 119902) 119892 (119905 119902)]
(49)
The scaling properties can be stated as follows
W119899119902[119891 (119886119905 119902)]
1003816100381610038161003816119905=0
= diag [1 119886 119886119899minus1]W119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0
(50)
P119899119902[119891 (119886119905 119902)]
1003816100381610038161003816119905=0
= diag [1 119886 119886119899minus1]P119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0
times diag [1 119886minus1 1198861minus119899]
(51)
Proof The formula (49) is proved by the 119902-Leibniz theoremFor (50) use the chain rule
Definition 16 Assume that 119891(0) = 0 If the inverse (119891(119905)minus1)(119896)exists for 119896 lt 119899 we can define the 119902-inverse of the generalized119902-Pascal functional matrix as
P119899119902[119891 (119905 119902)]
minus1119902
equiv P119899119902[119891(119905 119902)
minus1
] (52)
Example 17 (a 119902-analogue of [8 pages 231ndash234]) Let
(G119899119902) [119909] (119894 119895)
equiv
119892119894minus119895(119909) (
119894
119895)
119902
if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(53)
Put
119891 (119905 119909 119902) equiv
119899
sum
119896=0
119892119896(119909)
119905119896
119896119902 (54)
a truncated 119902-exponential generating function of 119902-binomialtype We find that
P119899119902[119891 (119905 119909 119902)]
1003816100381610038161003816119905=0 = G119899119902[119909] (55)
By a routine computation we obtain
G119899119902[119909]G119899119902[119910] = G
119899119902[119909 oplus119902119910] (56)
The function argument 119909oplus119902119910 is induced from the corre-
sponding 119902-additionIn the same way we obtain
G119899119902[119909]G1198991119902
[119910] = G119899119902[119909 ⊞119902119910] (57)
By the last formula we can define the inverse ofG119899119902[119909] as
G119899119902[119909]minus1
equiv G1198991119902
[minus119909] (58)
We now return permanently to [9]
Definition 18 (compare with [9 page 71]) LetΦ119902(119909) be the 119902-
Appell polynomial corresponding to 119891(119905) in (11) The vectorof 119902-Appell polynomials Φ
119902(119909) for 119891(119905) is given by
Φ119902(119909) (119894) equiv Φ
119894119902(119909) 119894 = 0 1 119899 minus 1 (59)
Theorem 19 (a 119902-analogue of [9 page 69]) Let 119891(119905) beanalytic in the neighbourhood of 0 with 119891(0) = 0 The 119902-difference equation in R119899
(D119902119909) 119910 (119909) = P
119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0 119910 (119909) minusinfin lt 119909 lt infin
(60)
with initial value 119910(0) = 1199100 has the following solution
119910 (119909) = P119899119902[E119902(119891 (119905) 119909)]
10038161003816100381610038161003816119905=01199100 (61)
Proof Since E119902(119860119905)|119905=0
is equal to the unit matrix the solu-tion of (60) is
Φ119902(119909) = E
119902(P119899119902[119891 (119905)]
1003816100381610038161003816119905=0119909) 1199100
= (
infin
sum
119896=0
P119896
119899119902[119891 (119905)])
1003816100381610038161003816100381610038161003816100381610038161003816119905=0
119909119896
119896119902
(62)
6 Journal of Discrete Mathematics
By (47) and (48)
Φ119902(119909) = P
119899119902(
infin
sum
119896=0
119891119896
(119905)119909119896
119896119902)
1003816100381610038161003816100381610038161003816100381610038161003816119905=0
(63)
where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot
120598
Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if
Φ (119909) = P119899119902[119891 (119905)]
1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(64)
Proof We know from [2 17] thatP119899119902[119905]|119905=0
= 119867119899119902 (65)
We thus obtainΦ (119909) = P
119899119902[E119902(119909119905)]
10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]
1003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(66)
Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by
P119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0 (67)
Can be compared with (53)
Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)
Ξ120584119902(119909) andΨ
120584119902(119909) be the 119902-Appell polynomials corresponding
to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then
Ξ120584119902(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119896119902(119911) (68)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)] sdot
120598W119899119902[ℎ (119905)E
119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 oplus119902119911) 119905)]
10038161003816100381610038161003816119905=0
(69)
By (11) the left-hand side of (69) is equal to
(((((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))))
)
times(
Ψ0119902(119911)
Ψ1119902(119911)
Ψ119899minus1119902
(119911)
)
(70)
By (11) the right-hand side of (69) is equal to
(
Ξ0119902(119910 oplus119902119911)
Ξ1119902(119910 oplus119902119911)
Ξ119899minus1119902
(119910 oplus119902119911)
) (71)
The theorem follows by equating the last rows of formulas (71)and (70)
Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which
enables inverses
Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902
and Ξ120584119902
be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902
120584be the pseudo 119902-Appell polynomial
for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then
Ξ120584119902(119910 ⊞119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119902
119896(119911) (72)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)]
10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E
1119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 ⊞119902119911) 119905)] |
119905=0
(73)
Journal of Discrete Mathematics 7
By (11) and (13) the left-hand side of (73) is equal to
(((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))
)
times(
Ψ119902
0(119911)
Ψ119902
1(119911)
Ψ119902
119899minus1(119911)
)
(74)
By (11) the right-hand side of (73) is equal to
(
Ξ0119902(119910 ⊞119902119911)
Ξ1119902(119910 ⊞119902119911)
Ξ119899minus1119902
(119910 ⊞119902119911)
) (75)
The theorem follows by equating the last rows of formulas(75) and (74)
The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)
Corollary 24 ([11 4242]) When sum119904119897=1
119899119897= 119899
B(119899)119873119882119860119896119902
(1199091oplus119902sdot sdot sdot oplus119902119909119904)
= sum
1198981+sdotsdotsdot+119898
119904=119896
(119896
1198981 119898
119904
)
119902
119904
prod
119895=1
B(119899119895)119873119882119860119898
119895119902(119909119895)
(76)
where we assume that 119899119895operates on 119909
119895
Yang and Youn have found a new identity betweenBernoulli and Euler polynomials
2119899B119899(119910 + 119911
2)
120584
sum
119896=0
(120584
119896)B119896(119910)E119899minus119896
(119911) (77)
The 119902-analogue looks as follows
Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider
L119873119882119860120584119902
(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
B119873119882119860120584minus119896119902
(119910) F119873119882119860120584119902
(119911)
(78)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E
119902(119905)+1)
in (68) Then
119892119899(119905) = 119891 (119905) 119892 (119905) =
(2119905)1
(E119902(1199052119902) minus 1)
(79)
This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers
Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)
be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-
mial Then119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(minus119909) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(80)
Let Φ119899119902(119909) and Ψ
119899119902(119909) be 119902-Appell polynomials Then
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910)Ψ119899minus119896119902
(119911) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902
(81)
Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E
119902(119905(119910 oplus
119902119911)) and ℎ(119905)
Or use an equivalent umbral reasoning
Corollary 27 One has the following special cases119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119909)B119902119873119882119860119899minus119896
(minus119909)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
B119873119882119860119899minus119896119902
(82)
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(83)
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(84)
8 Journal of Discrete Mathematics
Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ
119899119902(119909) and the pseudo-119902-Appell polynomial
Ψ119902
119899(119909) Then119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(119909) =
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(85)
Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and
Ψ119902
119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula
(50)
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0
=((
(
Φ0119902(119909) 0 sdot sdot sdot 0
Φ1119902(119909) Φ
0119902(119909) sdot sdot sdot 0
Φ2119902(119909) 2
119902Φ1119902(119909) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909) (119899 minus 1
1)
119902
Φ119899minus2119902
(119909) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909)
))
)
(
Ψ119902
0(119909)
minusΨ119902
1(119909)
(minus1)119899+1
Ψ119902
119899minus1(119909)
)
(86)
On the other hand by formula (49) we obtain
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot
120598W119899119902[ℎ (minus119905)]|
119905=0
=((
(
Φ0119902(0) 0 sdot sdot sdot 0
Φ1119902(0) Φ
0119902(0) sdot sdot sdot 0
Φ2119902(0) 2
119902Φ1119902(0) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(0) (119899 minus 1
1)
119902
Φ119899minus2119902
(0) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(0)
))
)
(
Ψ119902
0(0)
minusΨ119902
1(0)
(minus1)119899+1
Ψ119902
119899minus1(0)
)
(87)
The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and
(87)
We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902 (119909)B119902
NWA119899minus119896 (119909)
=
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902BNWA119899minus119896119902
(88)
Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ
119902(119909) and Ψ
119902(119909) be the 119902-Appell
vectors corresponding to 119891(119905) and 119892(119905) respectively Then
Φ119902(119909) = P
119899119902[119891 (119905)
119892 (119905)]
10038161003816100381610038161003816100381610038161003816119905=0
Ψ119902(119909) (89)
Proof By formula (49) we get
Φ119902(119909) = W
119899119902[119891 (119905)
119892 (119905)
119892 (119905)E119902(119909119905)]
10038161003816100381610038161003816100381610038161003816119905=0
= P119899119902[119891 (119905)
119892 (119905)] sdot120598W119899119902[119892 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(90)
Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials
B119873119882119860119899119902
(119909) =119899119902F119873119882119860119899minus1119902
(119909)
2
+
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119899minus119896119902
F119873119882119860119896119902
(119909)
(91)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E
119902(119905)+1)
in (89) Then
119891 (119905)
119892 (119905)=1199051
2+
1199051
E119902(119905) minus 1
(92)
Journal of Discrete Mathematics 9
We find that
(119891 (119905)
119892 (119905))
(119896)
(0) =
1
2+ BNWA119896119902 if 119896 = 1
BNWA119896119902 otherwise
(93)
The result follows by (89)
Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the
119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies
the following 119902-difference equation
119899
sum
119896=2
119902119899
120572119896minus1
119896 minus 1119902D119896119902119891 (119909119902
minus1
)
+ 119902119899
(119909119902 + 1205720)D119902119891 (119909119902
minus1
)
minus 119899 + 1119902119891 (119909) + 119902
119899
119891 (119909119902minus1
) = 0
(94)
where 120572119896equiv D119896119902(minus119891(119905)D
119902(1119891(119905)))|
119905=0
Proof We put 119891(119905) = 1119892(119905)
(
Φ1119902(119909)
Φ2119902(119909)
Φ119899119902(119909)
) = W119899119902[D119902(E119902(119909119905)
119892 (119905))]
100381610038161003816100381610038161003816100381610038161003816119905=0
= W119899119902[(119909 minus
D119902119892 (119905)
119892 (119905))E119902(119909119905)
119892 (119902119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
by (50)= P
119899119902[E119902(119909119905)
119892 (119902119905)] sdot120598W119899119902[119909 minus
D119902119892 (119905)
119892 (119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
=(((
(
Φ0119902(119909119902minus1
) 0 sdot sdot sdot 0
119902Φ1119902(119909119902minus1
) Φ0119902(119909119902minus1
) sdot sdot sdot 0
1199022
Φ2119902(119909119902minus1
) 1199022119902Φ1119902(119909119902minus1
) sdot sdot sdot 0
119902119899minus1
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909119902minus1
) 119902119899minus2
(119899 minus 1
1)
119902
Φ119899minus2119902
(119909119902minus1
) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909119902minus1
)
)))
)
(
119909+ 1205720
1205721
120572119899minus1
)
(95)
By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)
Φ120584119902(119909) = (119909 + 120572
0) 119902120584minus1
Φ120584minus1119902
(119909119902minus1
)
+
120584minus1
sum
119896=1
(120584 minus 1
119896)
119902
120572119896119902120584minus119896minus1
Φ120584minus119896minus1119902
(119909119902minus1
)
(96)
Operating with D119902on this and putting 120584 minus 1 = 119899
119899 + 1119902Φ119899119902(119909)
= 119902119899
Φ119899119902(119909119902minus1
)
+ 119902119899minus1
119899119902(119902119909 + 120572
0)Φ119899minus1119902
(119909119902minus1
)
+
119899
sum
119896=1
(119899
119896)
119902
120572119896119902119899minus119896minus1
119899 minus 119896119902
times Φ119899minus119896minus1119902
(119909119902minus1
) 119899 = 0 1 119899
(97)
where we have used the following identities
D119896119902Φ119899119902(119909119902minus1
) = 119902minus119896
119899 minus 119896 + 1119896119902Φ119899minus119896119902
D119899119902Φ119899minus1119902
(119909119902minus1
) = 0
(98)
4 Conclusion
By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]
10 Journal of Discrete Mathematics
Appendix
Bernoulli Number Formulas in Hansen
In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows
119899
sum
119896=0
(119899
119896)B119896(119910)B119899minus119896
(119911)
=
119899
sum
119896=0
(119899
119896)B119896(119910 + 119911)B
119899minus119896
(A1)
Hansen [10 page 341 50112] writes this as follows
119899
sum
119896=0
(minus1)119896(minus119899)119896
119896B119896(119910)B119899minus119896
(119911)
= 119899 (119910 + 119911 minus 1)B119899minus1
(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)
(A2)
Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials
References
[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012
[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted
[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963
[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923
[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009
[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967
[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006
[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007
[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009
[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser
2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169
Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981
[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951
[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition
[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009
[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted
[18] N E Noslashrlund Differenzenrechnung Springer 1924
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Discrete Mathematics
Definition 1 Let the Gauss 119902-binomial coefficient be definedby
(119899
119896)
119902
equiv119899119902
119896119902119899 minus 119896
119902 119896 = 0 1 119899 (1)
Let 119886 and 119887 be any elements with commutativemultiplicationThen the NWA 119902-addition is given by
(119886 oplus119902119887)119899
equiv
119899
sum
119896=0
(119899
119896)
119902
119886119896
119887119899minus119896
119899 = 0 1 2 (2)
The JHC 119902-addition is the function
(119886 ⊞119902119887)119899
equiv
119899
sum
119896=0
(119899
119896)
119902
119902(119896
2)
119887119896
119886119899minus119896
= 119886119899
(minus119887
119886 119902)
119899
119899 = 0 1 2
(3)
The 119902-exponential function is defined by
E119902(119911) equiv
infin
sum
119896=0
1
119896119902119911119896
(4)
The 119902-derivative is defined by
(D119902119891) (119909) equiv
119891 (119909) minus 119891 (119902119909)
(1 minus 119902) 119909 if 119902 isin C 1 119909 = 0
119889119891
119889119909(119909) if 119902 = 1
119889119891
119889119909(0) if 119909 = 0
(5)
Assume that 119905119910 and 119902 are the variables If wewant to indicatethe variable on which the 119902-difference operator is applied wedenote the operator in (5) by (D
119902119905120593)(119905 119910) To save space we
sometimes write 1198911015840 instead of D119902119905119891
Theorem2 The following equation relates the 119899th 119902-differenceoperator at zero to the 119899th derivative at zero for an analyticfunction 119891(119911) [12]
D119899119902119891 (0) =
119899119902
119899119891(119899)
(0) (6)
For convenience we will use the following two abbrevia-tions for functions 119891(119905) 119891(119896) represents the 119896th 119902-derivativeof 119891 and 119891
119896 represents the 119896th power of 119891 We will onlyconsider functions in C[[119905]] Sometimes functions can begiven in other forms we then mean the Taylor expansion ofthe function around zero In the sense ofMilne-Thomson [13]and Rainville [14] we write = to indicate the symbolic nature
If 119891(119909) isin C[119909] the ring of polynomials with complexcoefficients then the function 120598 C[119909] 997891rarr C[119909] is definedby
120598119891 (119909) equiv 119891 (119902119909) (7)
We make the convention that all 119899 times 119899 matrices are denotedby the first index 119899 The rows and columns are denoted by 119894and 119895 The 119902-Pascal matrix [2] is defined by
119875119899119902(119909) (119894 119895) equiv (
119894
119895)
119902
119909119894minus119895
(8)
The Polya-Vein matrix119867119899119902
[2] is given by
119867119899119902(119894 119894 minus 1) equiv 119894
119902 119894 = 1 119899 minus 1
119867119899119902(119894 119895) equiv 0 119895 = 119894 minus 1
(9)
2 119902-Appell Polynomials
The Appell polynomials have been around for more than acentury but this general notion is still not so well knownTheusual definition is
119891 (119905) 119890119909119905
=
infin
sum
120584=0
119905120584
120584Φ120584(119909) 119891 (119905) isin C [[119909]] (10)
The original article by Appell [15] started with the definition119889119860119899119889119909 = 119899119860
119899minus1 Then Appell showed that multiplication of
two Appell polynomials is commutative He then consideredthe inverse of119860
119899as a natural generalization of ordinary divi-
sionThe 119902-Appell polynomials form a natural generalizationof the Appell polynomials
21 Definitions Weconsider a certain value of the order (= 1)for the 119902-Appell polynomials in [11] The degree is only aname it is not the degree of polynomials
Definition 3 The 119902-Appell polynomials Φ120584119902(119909) of degree 120584
are defined by
119891 (119905)E119902(119909119905) =
infin
sum
120584=0
119905120584
120584119902Φ120584119902(119909) 119891 (119905) isin C [[119909]] (11)
We say that Φ120584119902(119909) is the 119902-Appell polynomial sequence for
119891(119905)We put 119909 = 0 and obtain
119891 (119905) =
infin
sum
120584=0
119905120584
120584119902Φ120584119902 Φ
0119902= 1 (12)
where Φ120584119902
is said to have degree 120584
Definition 4 The pseudo 119902-Appell polynomials Φ119902
120584(119909) of
degree 120584 [5] are defined by
119891 (119905)E1119902
(119909119905) =
infin
sum
120584=0
119905120584
120584119902Φ119902
120584(119909) (13)
We say that Φ119902120584(119909) is the pseudo 119902-Appell polynomial seq-
uence for 119891(119905)By putting 119909 = 0 we have
119891 (119905) =
infin
sum
120584=0
119905120584
120584119902Φ119902
120584 (14)
where Φ119902120584is called a Φ119902 number of degree 120584
Journal of Discrete Mathematics 3
We have similarly
D119902Φ119902
120584(119909) = 120584
119902Φ119902
120584minus1119902(119902119909)
Φ119902
120584(119909) = (Φ
119902⊞119902119909)120584
(15)
22 119902-Analogue of Carlitzrsquo Product Formula (1963) To moti-vate the generalized 119902-Pascal functional matrix we show thatthe Carlitz [3 page 247] formula for the product of generatingfunctions for Appell polynomials has a 119902-analogue involvingthe NWA 119902-addition Define the numbers Φ1015840
120584119902infin
120584=0by the
inverse of (12) (this is allowable since 119891(0) = 1)
119891(119905)minus1
=
infin
sum
120584=0
119905120584
120584119902Φ1015840
120584119902 Φ1015840
0119902= 1 (16)
Then we have the following
Theorem 5 An inverse relation formula for the 119902-Appellpolynomials
119909119899
=
119899
sum
119896=0
(119899
119896)
119902
Φ1015840
119899minus119896119902Φ119896119902(119909) (17)
Proof By the umbral definition of 119902-Appell polynomials wehave
119899
sum
119896=0
(119899
119896)
119902
Φ1015840
119899minus119896119902Φ119896119902(119909)
=
119899
sum
119896=0
(119899
119896)
119902
Φ1015840
119899minus119896119902
119896
sum
119897=0
(119896
119897)
119902
Φ119896minus119897119902
119909119897
=
119899
sum
119896=0
119899119902Φ1015840
119899minus119896119902
119899 minus 119896119902
119899
sum
119897=0
Φ119896minus119897119902
119897119902119896 minus 119897
119902119909119897
=
119899
sum
119897=0
119899119902
119897119902119909119897
sum
119904+119905=119899minus119897
Φ1015840
119904119902Φ119905119902
119904119902119905119902
=
119899
sum
119897=0
119899119902
119897119902119909119897
120575119899minus119897
= 119909119899
(18)
This completes the proof
We introduce two other 119902-Appell polynomials Ξ119902and Ψ
119902
by
119892 (119905)E119902(119909119905) =
infin
sum
120584=0
119905120584
120584119902Ξ120584119902(119909) 119892 (119905) isin C [[119909]]
119892 (119905) =
infin
sum
120584=0
119905120584
120584119902Ξ120584119902 Ξ0119902
= 1
ℎ (119905)E119902(119909119905) =
infin
sum
120584=0
119905120584
120584119902Ψ120584119902(119909) ℎ (119905) isin C [[119909]]
ℎ (119905) =
infin
sum
120584=0
119905120584
120584119902Ψ120584119902 Ψ0119902
= 1
(19)
Let us consider119891119898(119886119909) 119892
119899(119887119909)
=
119898
sum
119903=0
119899
sum
119904=0
(119898
119903)
119902
(119899
119904)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
119909119903+119904
(20)
Now put
ℎ(119905)minus1
=
infin
sum
120584=0
119905120584
120584119902Ψ1015840
120584119902 (21)
Because of (17) we get
119891119898(119886119909) 119892
119899(119887119909)
=
119898
sum
119903=0
119899
sum
119904=0
(119898
119903)
119902
(119899
119904)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
119903+119904
sum
119895=0
(119903 + 119904
119895)
119902
Ψ1015840
119903+119904minus119895119902Ψ119895119902(119909)
(22)
Introduce the notation119860 (119895 119902119898 119899 119886 119887)
equiv
119898
sum
119903=0
119899
sum
119904=0
(119898
119903)
119902
(119899
119904)
119902
(119903 + 119904
119895)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
Ψ1015840
119903+119904minus119895119902
(23)
where
119891119898(119886119909) 119892
119899(119887119909) =
119898+119899
sum
119895=0
119860 (119895 119902119898 119899 119886 119887)Ψ119895119902(119909) (24)
Theorem 6 With the previously mentioned notation one hasthe following 119902-analogue of Carlitz [3 page 247]
infin
sum
119898=0
infin
sum
119899=0
119860 (119895 119902119898 119899 119886 119887)119906119898
119907119899
119898119902119899119902
= 119891 (119906) 119892 (119907)
infin
sum
119896=0
Ψ1015840
119896119902
(119886119906 oplus119902119887119907)119896+119895
119895119902119896119902
(25)
Proof Thus we haveinfin
sum
119898=0
infin
sum
119899=0
119860 (119895 119902119898 119899 119886 119887)119906119898
119907119899
119898119902119899119902
=
infin
sum
119898=0
infin
sum
119899=0
119906119898
119907119899
119898119902119899119902
119898
sum
119903=0
119899
sum
119904=0
= (119898
119903)
119902
(119899
119904)
119902
(119903 + 119904
119895)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
Ψ1015840
119903+119904minus119895119902
=
infin
sum
119903=0
infin
sum
119904=0
(119886119906)119903
(119887119907)119904
119903119902119904119902
(119903 + 119904
119895)
119902
Ψ1015840
119903+119904minus119895119902
infin
sum
119898=0
infin
sum
119899=0
119891119898119892119899
119906119898
119907119899
119898119902119899119902
= 119891 (119906) 119892 (119907)
infin
sum
119903=0
infin
sum
119904=0
(119886119906)119903
(119887119907)119904
119903119902119904119902
4 Journal of Discrete Mathematics
(119903 + 119904
119895)
119902
Ψ1015840
119903+119904minus119895119902
= 119891 (119906) 119892 (119907)
infin
sum
119896=119895
(119896
119895)
119902
Ψ1015840
119896minus119895119902sum
119903+119904=119896
(119886119906)119903
(119887119907)119904
119903119902119904119902
= 119891 (119906) 119892 (119907)
infin
sum
119896=119895
(119896
119895)
119902
Ψ1015840
119896minus119895119902
(119886119906 oplus119902119887119907)119896
119896119902
(26)
23 Characterization of 119902-Appell Numbers We will now pre-sent a characterization of 119902-Appell numbers comparewithAl-Salam [6]
Definition 7 We denote by A119902the set of all 119902-Appell-
numbersLet Φ
119899119902and Ψ
119899119902be two elements in A
119902 Then the
operations + and ⋆ are defined as follows
(Φ119902+ Ψ119902)119899
equiv Φ119899119902+ Ψ119899119902 (27)
(Φ119902⋆ Ψ119902)119899
equiv
119899
sum
119896=0
(119899
119896)
119902
Φ119899minus119896119902
Ψ119896119902 (28)
Theorem 8 The identity element in A119902with respect to ⋆ is
119868 equiv 1205751198990 This corresponds to 119891(119905) = 1
Proof Put 119896 = 119899 and Φ119902= 1 in (28)
Theorem 9 (Compare with [6 page 34]) Let Φ119899119902 Ξ119899119902
andΨ119899119902
be three elements in A119902corresponding to the generating
functions119891(119905) 119892(119905) and ℎ(119905) respectively Further assume that119891(0) + ℎ(0) = 0 Then
Φ119899119902+ Ψ119899119902
isin A119902
Φ119899119902+ Ψ119899119902
has generating function119891 (119905) + ℎ (119905) (29)
The associative law for addition reads
Φ119899119902+ (Ξ119899119902+ Ψ119899119902) = (Φ
119899119902+ Ξ119899119902) + Ψ119899119902 (30)
Proof This follows from the definitions
Theorem 10 (Compare with [6 page 34]) LetΦ119899119902 Ξ119899119902 and
Ψ119899119902
be three elements in A119902corresponding to the generating
functions 119891(119905) 119892(119905) and ℎ(119905) respectively Then
Φ119899119902⋆ Ψ119899119902
isin A119902 (31)
Φ119899119902⋆ Ψ119899119902
= Ψ119899119902⋆ Φ119899119902 (32)
Φ119899119902⋆ Ψ119899119902
has generating function119891 (119905) ℎ (119905) (33)
The associative law for ⋆ reads
Φ119899119902⋆ (Ξ119899119902⋆ Ψ119899119902) = (Φ
119899119902⋆ Ξ119899119902) ⋆ Ψ119899119902 (34)
Proof We prove (31) and (33) at the same time We have inthe umbral sense
119891 (119905) ℎ (119905) = E119902(119905 (Φ119902oplus119902Ψ119902)) (35)
Formulas (32) and (34) follow by the commutativity andthe associativity of NWA
24 Specializations
Definition 11 The NWA 119902-Bernoulli polynomialsBNWA120584119902(119909) of degree 120584 are defined by
1199051
(E119902(119905) minus 1)
E119902(119909119905) =
infin
sum
120584=0
119905120584BNWA120584119902 (119909)
120584119902
|119905| lt 2120587 (36)
The NWA 119902-Bernoulli polynomials are also given by
BNWA120584119902 (119909) = (BNWA119902 oplus119902 119909)120584
(37)
The pseudo-119902-Bernoulli polynomials are defined by
B119902NWA120584 (119909) = (BNWA119902 ⊞119902 119909)120584
(38)
The JHC 119902-Bernoulli polynomials BJHC120584119902(119909) of degree 120584are defined by
1199051
(E1119902
(119905) minus 1)E119902(119909119905) =
infin
sum
120584=0
119905120584BJHC120584119902 (119909)
120584119902
|119905| lt 2120587 (39)
3 119902-Pascal and 119902-Wronskian Matrices
We now come to the most important part of the paper whichis of interest even for the case 119902 = 1 since the original paperby Yang and Youn is not very widespread
Definition 12 The generalized 119902-Pascal functional matrix isgiven by
(P119899119902) [119891 (119905 119902)] (119894 119895)
equiv
(119894
119895)
119902
D119894minus119895119902119905119891 (119905 119902) if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(40)
In the special case 119891(119905 119902) = constant
(P119899119902) [119891 (119905 119902)] = 119891 (119905 119902) 119868 (41)
Definition 13 The 119902-deformed Wronskian vector is given by
(W119899119902) [119891 (119905 119902)] (119894 0) equiv D119894
119902119905119891 (119905 119902) 119894 = 0 1 119899 minus 1
(42)
Journal of Discrete Mathematics 5
Definition 14 The 119902-deformed Leibniz functional matrix isgiven by(L119899119902) [119891 (119905 119902)] (119894 119895)
equiv
D119894minus119895119902119905119891 (119905 119902)
119894 minus 119895119902
if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(43)
In order to find a 119902-analogue of the formula [16]
(L119899) [119891 (119905) 119892 (119905)] = (L
119899) [119891 (119905)] (L
119899) [119892 (119905)] (44)
we infer that by the 119902-Leibniz formula [1](L119899119902) [119891 (119905 119902) 119892 (119905 119902)]
= (L119899119902) [119891 (119905 119902)] sdot
120598(L119899119902) [119892 (119905 119902)]
(45)
where in the matrix multiplication for every term whichincludes D119896
119902119891 we operate with 120598119896 on 119892 We denote this by sdot
120598
This operator can also be iterated compare with [1]
Theorem 15 (a 119902-analogue of [9 page 67]) Consider
P119899119902[E119902(119909119905)]
10038161003816100381610038161003816119905=0= 119875119899119902(119909) (46)
where the 119902-Pascal matrix is defined in (8)The mappingsP
119899119902[sdot] andW
119899119902[sdot] are linear that is
P119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886P
119899119902[119891 (119905)] + 119887P
119899119902[119892 (119905)]
W119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886W
119899119902[119891 (119905)] + 119887W
119899119902[119892 (119905)]
(47)
One has the following two matrix multiplication formulasthe first one is a 119902-analogue of [8 page 232]
P119899119902[119891 (119905 119902)] sdot
120598P119899119902[119892 (119905 119902)] = P
119899119902[119891 (119905 119902) 119892 (119905 119902)]
(48)
P119899119902[119891 (119905 119902)] sdot
120598W119899119902[119892 (119905 119902)] = W
119899119902[119891 (119905 119902) 119892 (119905 119902)]
(49)
The scaling properties can be stated as follows
W119899119902[119891 (119886119905 119902)]
1003816100381610038161003816119905=0
= diag [1 119886 119886119899minus1]W119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0
(50)
P119899119902[119891 (119886119905 119902)]
1003816100381610038161003816119905=0
= diag [1 119886 119886119899minus1]P119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0
times diag [1 119886minus1 1198861minus119899]
(51)
Proof The formula (49) is proved by the 119902-Leibniz theoremFor (50) use the chain rule
Definition 16 Assume that 119891(0) = 0 If the inverse (119891(119905)minus1)(119896)exists for 119896 lt 119899 we can define the 119902-inverse of the generalized119902-Pascal functional matrix as
P119899119902[119891 (119905 119902)]
minus1119902
equiv P119899119902[119891(119905 119902)
minus1
] (52)
Example 17 (a 119902-analogue of [8 pages 231ndash234]) Let
(G119899119902) [119909] (119894 119895)
equiv
119892119894minus119895(119909) (
119894
119895)
119902
if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(53)
Put
119891 (119905 119909 119902) equiv
119899
sum
119896=0
119892119896(119909)
119905119896
119896119902 (54)
a truncated 119902-exponential generating function of 119902-binomialtype We find that
P119899119902[119891 (119905 119909 119902)]
1003816100381610038161003816119905=0 = G119899119902[119909] (55)
By a routine computation we obtain
G119899119902[119909]G119899119902[119910] = G
119899119902[119909 oplus119902119910] (56)
The function argument 119909oplus119902119910 is induced from the corre-
sponding 119902-additionIn the same way we obtain
G119899119902[119909]G1198991119902
[119910] = G119899119902[119909 ⊞119902119910] (57)
By the last formula we can define the inverse ofG119899119902[119909] as
G119899119902[119909]minus1
equiv G1198991119902
[minus119909] (58)
We now return permanently to [9]
Definition 18 (compare with [9 page 71]) LetΦ119902(119909) be the 119902-
Appell polynomial corresponding to 119891(119905) in (11) The vectorof 119902-Appell polynomials Φ
119902(119909) for 119891(119905) is given by
Φ119902(119909) (119894) equiv Φ
119894119902(119909) 119894 = 0 1 119899 minus 1 (59)
Theorem 19 (a 119902-analogue of [9 page 69]) Let 119891(119905) beanalytic in the neighbourhood of 0 with 119891(0) = 0 The 119902-difference equation in R119899
(D119902119909) 119910 (119909) = P
119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0 119910 (119909) minusinfin lt 119909 lt infin
(60)
with initial value 119910(0) = 1199100 has the following solution
119910 (119909) = P119899119902[E119902(119891 (119905) 119909)]
10038161003816100381610038161003816119905=01199100 (61)
Proof Since E119902(119860119905)|119905=0
is equal to the unit matrix the solu-tion of (60) is
Φ119902(119909) = E
119902(P119899119902[119891 (119905)]
1003816100381610038161003816119905=0119909) 1199100
= (
infin
sum
119896=0
P119896
119899119902[119891 (119905)])
1003816100381610038161003816100381610038161003816100381610038161003816119905=0
119909119896
119896119902
(62)
6 Journal of Discrete Mathematics
By (47) and (48)
Φ119902(119909) = P
119899119902(
infin
sum
119896=0
119891119896
(119905)119909119896
119896119902)
1003816100381610038161003816100381610038161003816100381610038161003816119905=0
(63)
where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot
120598
Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if
Φ (119909) = P119899119902[119891 (119905)]
1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(64)
Proof We know from [2 17] thatP119899119902[119905]|119905=0
= 119867119899119902 (65)
We thus obtainΦ (119909) = P
119899119902[E119902(119909119905)]
10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]
1003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(66)
Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by
P119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0 (67)
Can be compared with (53)
Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)
Ξ120584119902(119909) andΨ
120584119902(119909) be the 119902-Appell polynomials corresponding
to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then
Ξ120584119902(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119896119902(119911) (68)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)] sdot
120598W119899119902[ℎ (119905)E
119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 oplus119902119911) 119905)]
10038161003816100381610038161003816119905=0
(69)
By (11) the left-hand side of (69) is equal to
(((((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))))
)
times(
Ψ0119902(119911)
Ψ1119902(119911)
Ψ119899minus1119902
(119911)
)
(70)
By (11) the right-hand side of (69) is equal to
(
Ξ0119902(119910 oplus119902119911)
Ξ1119902(119910 oplus119902119911)
Ξ119899minus1119902
(119910 oplus119902119911)
) (71)
The theorem follows by equating the last rows of formulas (71)and (70)
Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which
enables inverses
Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902
and Ξ120584119902
be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902
120584be the pseudo 119902-Appell polynomial
for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then
Ξ120584119902(119910 ⊞119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119902
119896(119911) (72)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)]
10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E
1119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 ⊞119902119911) 119905)] |
119905=0
(73)
Journal of Discrete Mathematics 7
By (11) and (13) the left-hand side of (73) is equal to
(((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))
)
times(
Ψ119902
0(119911)
Ψ119902
1(119911)
Ψ119902
119899minus1(119911)
)
(74)
By (11) the right-hand side of (73) is equal to
(
Ξ0119902(119910 ⊞119902119911)
Ξ1119902(119910 ⊞119902119911)
Ξ119899minus1119902
(119910 ⊞119902119911)
) (75)
The theorem follows by equating the last rows of formulas(75) and (74)
The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)
Corollary 24 ([11 4242]) When sum119904119897=1
119899119897= 119899
B(119899)119873119882119860119896119902
(1199091oplus119902sdot sdot sdot oplus119902119909119904)
= sum
1198981+sdotsdotsdot+119898
119904=119896
(119896
1198981 119898
119904
)
119902
119904
prod
119895=1
B(119899119895)119873119882119860119898
119895119902(119909119895)
(76)
where we assume that 119899119895operates on 119909
119895
Yang and Youn have found a new identity betweenBernoulli and Euler polynomials
2119899B119899(119910 + 119911
2)
120584
sum
119896=0
(120584
119896)B119896(119910)E119899minus119896
(119911) (77)
The 119902-analogue looks as follows
Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider
L119873119882119860120584119902
(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
B119873119882119860120584minus119896119902
(119910) F119873119882119860120584119902
(119911)
(78)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E
119902(119905)+1)
in (68) Then
119892119899(119905) = 119891 (119905) 119892 (119905) =
(2119905)1
(E119902(1199052119902) minus 1)
(79)
This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers
Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)
be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-
mial Then119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(minus119909) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(80)
Let Φ119899119902(119909) and Ψ
119899119902(119909) be 119902-Appell polynomials Then
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910)Ψ119899minus119896119902
(119911) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902
(81)
Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E
119902(119905(119910 oplus
119902119911)) and ℎ(119905)
Or use an equivalent umbral reasoning
Corollary 27 One has the following special cases119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119909)B119902119873119882119860119899minus119896
(minus119909)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
B119873119882119860119899minus119896119902
(82)
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(83)
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(84)
8 Journal of Discrete Mathematics
Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ
119899119902(119909) and the pseudo-119902-Appell polynomial
Ψ119902
119899(119909) Then119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(119909) =
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(85)
Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and
Ψ119902
119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula
(50)
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0
=((
(
Φ0119902(119909) 0 sdot sdot sdot 0
Φ1119902(119909) Φ
0119902(119909) sdot sdot sdot 0
Φ2119902(119909) 2
119902Φ1119902(119909) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909) (119899 minus 1
1)
119902
Φ119899minus2119902
(119909) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909)
))
)
(
Ψ119902
0(119909)
minusΨ119902
1(119909)
(minus1)119899+1
Ψ119902
119899minus1(119909)
)
(86)
On the other hand by formula (49) we obtain
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot
120598W119899119902[ℎ (minus119905)]|
119905=0
=((
(
Φ0119902(0) 0 sdot sdot sdot 0
Φ1119902(0) Φ
0119902(0) sdot sdot sdot 0
Φ2119902(0) 2
119902Φ1119902(0) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(0) (119899 minus 1
1)
119902
Φ119899minus2119902
(0) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(0)
))
)
(
Ψ119902
0(0)
minusΨ119902
1(0)
(minus1)119899+1
Ψ119902
119899minus1(0)
)
(87)
The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and
(87)
We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902 (119909)B119902
NWA119899minus119896 (119909)
=
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902BNWA119899minus119896119902
(88)
Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ
119902(119909) and Ψ
119902(119909) be the 119902-Appell
vectors corresponding to 119891(119905) and 119892(119905) respectively Then
Φ119902(119909) = P
119899119902[119891 (119905)
119892 (119905)]
10038161003816100381610038161003816100381610038161003816119905=0
Ψ119902(119909) (89)
Proof By formula (49) we get
Φ119902(119909) = W
119899119902[119891 (119905)
119892 (119905)
119892 (119905)E119902(119909119905)]
10038161003816100381610038161003816100381610038161003816119905=0
= P119899119902[119891 (119905)
119892 (119905)] sdot120598W119899119902[119892 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(90)
Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials
B119873119882119860119899119902
(119909) =119899119902F119873119882119860119899minus1119902
(119909)
2
+
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119899minus119896119902
F119873119882119860119896119902
(119909)
(91)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E
119902(119905)+1)
in (89) Then
119891 (119905)
119892 (119905)=1199051
2+
1199051
E119902(119905) minus 1
(92)
Journal of Discrete Mathematics 9
We find that
(119891 (119905)
119892 (119905))
(119896)
(0) =
1
2+ BNWA119896119902 if 119896 = 1
BNWA119896119902 otherwise
(93)
The result follows by (89)
Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the
119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies
the following 119902-difference equation
119899
sum
119896=2
119902119899
120572119896minus1
119896 minus 1119902D119896119902119891 (119909119902
minus1
)
+ 119902119899
(119909119902 + 1205720)D119902119891 (119909119902
minus1
)
minus 119899 + 1119902119891 (119909) + 119902
119899
119891 (119909119902minus1
) = 0
(94)
where 120572119896equiv D119896119902(minus119891(119905)D
119902(1119891(119905)))|
119905=0
Proof We put 119891(119905) = 1119892(119905)
(
Φ1119902(119909)
Φ2119902(119909)
Φ119899119902(119909)
) = W119899119902[D119902(E119902(119909119905)
119892 (119905))]
100381610038161003816100381610038161003816100381610038161003816119905=0
= W119899119902[(119909 minus
D119902119892 (119905)
119892 (119905))E119902(119909119905)
119892 (119902119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
by (50)= P
119899119902[E119902(119909119905)
119892 (119902119905)] sdot120598W119899119902[119909 minus
D119902119892 (119905)
119892 (119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
=(((
(
Φ0119902(119909119902minus1
) 0 sdot sdot sdot 0
119902Φ1119902(119909119902minus1
) Φ0119902(119909119902minus1
) sdot sdot sdot 0
1199022
Φ2119902(119909119902minus1
) 1199022119902Φ1119902(119909119902minus1
) sdot sdot sdot 0
119902119899minus1
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909119902minus1
) 119902119899minus2
(119899 minus 1
1)
119902
Φ119899minus2119902
(119909119902minus1
) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909119902minus1
)
)))
)
(
119909+ 1205720
1205721
120572119899minus1
)
(95)
By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)
Φ120584119902(119909) = (119909 + 120572
0) 119902120584minus1
Φ120584minus1119902
(119909119902minus1
)
+
120584minus1
sum
119896=1
(120584 minus 1
119896)
119902
120572119896119902120584minus119896minus1
Φ120584minus119896minus1119902
(119909119902minus1
)
(96)
Operating with D119902on this and putting 120584 minus 1 = 119899
119899 + 1119902Φ119899119902(119909)
= 119902119899
Φ119899119902(119909119902minus1
)
+ 119902119899minus1
119899119902(119902119909 + 120572
0)Φ119899minus1119902
(119909119902minus1
)
+
119899
sum
119896=1
(119899
119896)
119902
120572119896119902119899minus119896minus1
119899 minus 119896119902
times Φ119899minus119896minus1119902
(119909119902minus1
) 119899 = 0 1 119899
(97)
where we have used the following identities
D119896119902Φ119899119902(119909119902minus1
) = 119902minus119896
119899 minus 119896 + 1119896119902Φ119899minus119896119902
D119899119902Φ119899minus1119902
(119909119902minus1
) = 0
(98)
4 Conclusion
By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]
10 Journal of Discrete Mathematics
Appendix
Bernoulli Number Formulas in Hansen
In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows
119899
sum
119896=0
(119899
119896)B119896(119910)B119899minus119896
(119911)
=
119899
sum
119896=0
(119899
119896)B119896(119910 + 119911)B
119899minus119896
(A1)
Hansen [10 page 341 50112] writes this as follows
119899
sum
119896=0
(minus1)119896(minus119899)119896
119896B119896(119910)B119899minus119896
(119911)
= 119899 (119910 + 119911 minus 1)B119899minus1
(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)
(A2)
Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials
References
[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012
[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted
[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963
[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923
[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009
[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967
[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006
[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007
[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009
[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser
2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169
Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981
[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951
[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition
[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009
[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted
[18] N E Noslashrlund Differenzenrechnung Springer 1924
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Discrete Mathematics 3
We have similarly
D119902Φ119902
120584(119909) = 120584
119902Φ119902
120584minus1119902(119902119909)
Φ119902
120584(119909) = (Φ
119902⊞119902119909)120584
(15)
22 119902-Analogue of Carlitzrsquo Product Formula (1963) To moti-vate the generalized 119902-Pascal functional matrix we show thatthe Carlitz [3 page 247] formula for the product of generatingfunctions for Appell polynomials has a 119902-analogue involvingthe NWA 119902-addition Define the numbers Φ1015840
120584119902infin
120584=0by the
inverse of (12) (this is allowable since 119891(0) = 1)
119891(119905)minus1
=
infin
sum
120584=0
119905120584
120584119902Φ1015840
120584119902 Φ1015840
0119902= 1 (16)
Then we have the following
Theorem 5 An inverse relation formula for the 119902-Appellpolynomials
119909119899
=
119899
sum
119896=0
(119899
119896)
119902
Φ1015840
119899minus119896119902Φ119896119902(119909) (17)
Proof By the umbral definition of 119902-Appell polynomials wehave
119899
sum
119896=0
(119899
119896)
119902
Φ1015840
119899minus119896119902Φ119896119902(119909)
=
119899
sum
119896=0
(119899
119896)
119902
Φ1015840
119899minus119896119902
119896
sum
119897=0
(119896
119897)
119902
Φ119896minus119897119902
119909119897
=
119899
sum
119896=0
119899119902Φ1015840
119899minus119896119902
119899 minus 119896119902
119899
sum
119897=0
Φ119896minus119897119902
119897119902119896 minus 119897
119902119909119897
=
119899
sum
119897=0
119899119902
119897119902119909119897
sum
119904+119905=119899minus119897
Φ1015840
119904119902Φ119905119902
119904119902119905119902
=
119899
sum
119897=0
119899119902
119897119902119909119897
120575119899minus119897
= 119909119899
(18)
This completes the proof
We introduce two other 119902-Appell polynomials Ξ119902and Ψ
119902
by
119892 (119905)E119902(119909119905) =
infin
sum
120584=0
119905120584
120584119902Ξ120584119902(119909) 119892 (119905) isin C [[119909]]
119892 (119905) =
infin
sum
120584=0
119905120584
120584119902Ξ120584119902 Ξ0119902
= 1
ℎ (119905)E119902(119909119905) =
infin
sum
120584=0
119905120584
120584119902Ψ120584119902(119909) ℎ (119905) isin C [[119909]]
ℎ (119905) =
infin
sum
120584=0
119905120584
120584119902Ψ120584119902 Ψ0119902
= 1
(19)
Let us consider119891119898(119886119909) 119892
119899(119887119909)
=
119898
sum
119903=0
119899
sum
119904=0
(119898
119903)
119902
(119899
119904)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
119909119903+119904
(20)
Now put
ℎ(119905)minus1
=
infin
sum
120584=0
119905120584
120584119902Ψ1015840
120584119902 (21)
Because of (17) we get
119891119898(119886119909) 119892
119899(119887119909)
=
119898
sum
119903=0
119899
sum
119904=0
(119898
119903)
119902
(119899
119904)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
119903+119904
sum
119895=0
(119903 + 119904
119895)
119902
Ψ1015840
119903+119904minus119895119902Ψ119895119902(119909)
(22)
Introduce the notation119860 (119895 119902119898 119899 119886 119887)
equiv
119898
sum
119903=0
119899
sum
119904=0
(119898
119903)
119902
(119899
119904)
119902
(119903 + 119904
119895)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
Ψ1015840
119903+119904minus119895119902
(23)
where
119891119898(119886119909) 119892
119899(119887119909) =
119898+119899
sum
119895=0
119860 (119895 119902119898 119899 119886 119887)Ψ119895119902(119909) (24)
Theorem 6 With the previously mentioned notation one hasthe following 119902-analogue of Carlitz [3 page 247]
infin
sum
119898=0
infin
sum
119899=0
119860 (119895 119902119898 119899 119886 119887)119906119898
119907119899
119898119902119899119902
= 119891 (119906) 119892 (119907)
infin
sum
119896=0
Ψ1015840
119896119902
(119886119906 oplus119902119887119907)119896+119895
119895119902119896119902
(25)
Proof Thus we haveinfin
sum
119898=0
infin
sum
119899=0
119860 (119895 119902119898 119899 119886 119887)119906119898
119907119899
119898119902119899119902
=
infin
sum
119898=0
infin
sum
119899=0
119906119898
119907119899
119898119902119899119902
119898
sum
119903=0
119899
sum
119904=0
= (119898
119903)
119902
(119899
119904)
119902
(119903 + 119904
119895)
119902
119891119898minus119903
119892119899minus119904119886119903
119887119904
Ψ1015840
119903+119904minus119895119902
=
infin
sum
119903=0
infin
sum
119904=0
(119886119906)119903
(119887119907)119904
119903119902119904119902
(119903 + 119904
119895)
119902
Ψ1015840
119903+119904minus119895119902
infin
sum
119898=0
infin
sum
119899=0
119891119898119892119899
119906119898
119907119899
119898119902119899119902
= 119891 (119906) 119892 (119907)
infin
sum
119903=0
infin
sum
119904=0
(119886119906)119903
(119887119907)119904
119903119902119904119902
4 Journal of Discrete Mathematics
(119903 + 119904
119895)
119902
Ψ1015840
119903+119904minus119895119902
= 119891 (119906) 119892 (119907)
infin
sum
119896=119895
(119896
119895)
119902
Ψ1015840
119896minus119895119902sum
119903+119904=119896
(119886119906)119903
(119887119907)119904
119903119902119904119902
= 119891 (119906) 119892 (119907)
infin
sum
119896=119895
(119896
119895)
119902
Ψ1015840
119896minus119895119902
(119886119906 oplus119902119887119907)119896
119896119902
(26)
23 Characterization of 119902-Appell Numbers We will now pre-sent a characterization of 119902-Appell numbers comparewithAl-Salam [6]
Definition 7 We denote by A119902the set of all 119902-Appell-
numbersLet Φ
119899119902and Ψ
119899119902be two elements in A
119902 Then the
operations + and ⋆ are defined as follows
(Φ119902+ Ψ119902)119899
equiv Φ119899119902+ Ψ119899119902 (27)
(Φ119902⋆ Ψ119902)119899
equiv
119899
sum
119896=0
(119899
119896)
119902
Φ119899minus119896119902
Ψ119896119902 (28)
Theorem 8 The identity element in A119902with respect to ⋆ is
119868 equiv 1205751198990 This corresponds to 119891(119905) = 1
Proof Put 119896 = 119899 and Φ119902= 1 in (28)
Theorem 9 (Compare with [6 page 34]) Let Φ119899119902 Ξ119899119902
andΨ119899119902
be three elements in A119902corresponding to the generating
functions119891(119905) 119892(119905) and ℎ(119905) respectively Further assume that119891(0) + ℎ(0) = 0 Then
Φ119899119902+ Ψ119899119902
isin A119902
Φ119899119902+ Ψ119899119902
has generating function119891 (119905) + ℎ (119905) (29)
The associative law for addition reads
Φ119899119902+ (Ξ119899119902+ Ψ119899119902) = (Φ
119899119902+ Ξ119899119902) + Ψ119899119902 (30)
Proof This follows from the definitions
Theorem 10 (Compare with [6 page 34]) LetΦ119899119902 Ξ119899119902 and
Ψ119899119902
be three elements in A119902corresponding to the generating
functions 119891(119905) 119892(119905) and ℎ(119905) respectively Then
Φ119899119902⋆ Ψ119899119902
isin A119902 (31)
Φ119899119902⋆ Ψ119899119902
= Ψ119899119902⋆ Φ119899119902 (32)
Φ119899119902⋆ Ψ119899119902
has generating function119891 (119905) ℎ (119905) (33)
The associative law for ⋆ reads
Φ119899119902⋆ (Ξ119899119902⋆ Ψ119899119902) = (Φ
119899119902⋆ Ξ119899119902) ⋆ Ψ119899119902 (34)
Proof We prove (31) and (33) at the same time We have inthe umbral sense
119891 (119905) ℎ (119905) = E119902(119905 (Φ119902oplus119902Ψ119902)) (35)
Formulas (32) and (34) follow by the commutativity andthe associativity of NWA
24 Specializations
Definition 11 The NWA 119902-Bernoulli polynomialsBNWA120584119902(119909) of degree 120584 are defined by
1199051
(E119902(119905) minus 1)
E119902(119909119905) =
infin
sum
120584=0
119905120584BNWA120584119902 (119909)
120584119902
|119905| lt 2120587 (36)
The NWA 119902-Bernoulli polynomials are also given by
BNWA120584119902 (119909) = (BNWA119902 oplus119902 119909)120584
(37)
The pseudo-119902-Bernoulli polynomials are defined by
B119902NWA120584 (119909) = (BNWA119902 ⊞119902 119909)120584
(38)
The JHC 119902-Bernoulli polynomials BJHC120584119902(119909) of degree 120584are defined by
1199051
(E1119902
(119905) minus 1)E119902(119909119905) =
infin
sum
120584=0
119905120584BJHC120584119902 (119909)
120584119902
|119905| lt 2120587 (39)
3 119902-Pascal and 119902-Wronskian Matrices
We now come to the most important part of the paper whichis of interest even for the case 119902 = 1 since the original paperby Yang and Youn is not very widespread
Definition 12 The generalized 119902-Pascal functional matrix isgiven by
(P119899119902) [119891 (119905 119902)] (119894 119895)
equiv
(119894
119895)
119902
D119894minus119895119902119905119891 (119905 119902) if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(40)
In the special case 119891(119905 119902) = constant
(P119899119902) [119891 (119905 119902)] = 119891 (119905 119902) 119868 (41)
Definition 13 The 119902-deformed Wronskian vector is given by
(W119899119902) [119891 (119905 119902)] (119894 0) equiv D119894
119902119905119891 (119905 119902) 119894 = 0 1 119899 minus 1
(42)
Journal of Discrete Mathematics 5
Definition 14 The 119902-deformed Leibniz functional matrix isgiven by(L119899119902) [119891 (119905 119902)] (119894 119895)
equiv
D119894minus119895119902119905119891 (119905 119902)
119894 minus 119895119902
if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(43)
In order to find a 119902-analogue of the formula [16]
(L119899) [119891 (119905) 119892 (119905)] = (L
119899) [119891 (119905)] (L
119899) [119892 (119905)] (44)
we infer that by the 119902-Leibniz formula [1](L119899119902) [119891 (119905 119902) 119892 (119905 119902)]
= (L119899119902) [119891 (119905 119902)] sdot
120598(L119899119902) [119892 (119905 119902)]
(45)
where in the matrix multiplication for every term whichincludes D119896
119902119891 we operate with 120598119896 on 119892 We denote this by sdot
120598
This operator can also be iterated compare with [1]
Theorem 15 (a 119902-analogue of [9 page 67]) Consider
P119899119902[E119902(119909119905)]
10038161003816100381610038161003816119905=0= 119875119899119902(119909) (46)
where the 119902-Pascal matrix is defined in (8)The mappingsP
119899119902[sdot] andW
119899119902[sdot] are linear that is
P119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886P
119899119902[119891 (119905)] + 119887P
119899119902[119892 (119905)]
W119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886W
119899119902[119891 (119905)] + 119887W
119899119902[119892 (119905)]
(47)
One has the following two matrix multiplication formulasthe first one is a 119902-analogue of [8 page 232]
P119899119902[119891 (119905 119902)] sdot
120598P119899119902[119892 (119905 119902)] = P
119899119902[119891 (119905 119902) 119892 (119905 119902)]
(48)
P119899119902[119891 (119905 119902)] sdot
120598W119899119902[119892 (119905 119902)] = W
119899119902[119891 (119905 119902) 119892 (119905 119902)]
(49)
The scaling properties can be stated as follows
W119899119902[119891 (119886119905 119902)]
1003816100381610038161003816119905=0
= diag [1 119886 119886119899minus1]W119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0
(50)
P119899119902[119891 (119886119905 119902)]
1003816100381610038161003816119905=0
= diag [1 119886 119886119899minus1]P119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0
times diag [1 119886minus1 1198861minus119899]
(51)
Proof The formula (49) is proved by the 119902-Leibniz theoremFor (50) use the chain rule
Definition 16 Assume that 119891(0) = 0 If the inverse (119891(119905)minus1)(119896)exists for 119896 lt 119899 we can define the 119902-inverse of the generalized119902-Pascal functional matrix as
P119899119902[119891 (119905 119902)]
minus1119902
equiv P119899119902[119891(119905 119902)
minus1
] (52)
Example 17 (a 119902-analogue of [8 pages 231ndash234]) Let
(G119899119902) [119909] (119894 119895)
equiv
119892119894minus119895(119909) (
119894
119895)
119902
if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(53)
Put
119891 (119905 119909 119902) equiv
119899
sum
119896=0
119892119896(119909)
119905119896
119896119902 (54)
a truncated 119902-exponential generating function of 119902-binomialtype We find that
P119899119902[119891 (119905 119909 119902)]
1003816100381610038161003816119905=0 = G119899119902[119909] (55)
By a routine computation we obtain
G119899119902[119909]G119899119902[119910] = G
119899119902[119909 oplus119902119910] (56)
The function argument 119909oplus119902119910 is induced from the corre-
sponding 119902-additionIn the same way we obtain
G119899119902[119909]G1198991119902
[119910] = G119899119902[119909 ⊞119902119910] (57)
By the last formula we can define the inverse ofG119899119902[119909] as
G119899119902[119909]minus1
equiv G1198991119902
[minus119909] (58)
We now return permanently to [9]
Definition 18 (compare with [9 page 71]) LetΦ119902(119909) be the 119902-
Appell polynomial corresponding to 119891(119905) in (11) The vectorof 119902-Appell polynomials Φ
119902(119909) for 119891(119905) is given by
Φ119902(119909) (119894) equiv Φ
119894119902(119909) 119894 = 0 1 119899 minus 1 (59)
Theorem 19 (a 119902-analogue of [9 page 69]) Let 119891(119905) beanalytic in the neighbourhood of 0 with 119891(0) = 0 The 119902-difference equation in R119899
(D119902119909) 119910 (119909) = P
119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0 119910 (119909) minusinfin lt 119909 lt infin
(60)
with initial value 119910(0) = 1199100 has the following solution
119910 (119909) = P119899119902[E119902(119891 (119905) 119909)]
10038161003816100381610038161003816119905=01199100 (61)
Proof Since E119902(119860119905)|119905=0
is equal to the unit matrix the solu-tion of (60) is
Φ119902(119909) = E
119902(P119899119902[119891 (119905)]
1003816100381610038161003816119905=0119909) 1199100
= (
infin
sum
119896=0
P119896
119899119902[119891 (119905)])
1003816100381610038161003816100381610038161003816100381610038161003816119905=0
119909119896
119896119902
(62)
6 Journal of Discrete Mathematics
By (47) and (48)
Φ119902(119909) = P
119899119902(
infin
sum
119896=0
119891119896
(119905)119909119896
119896119902)
1003816100381610038161003816100381610038161003816100381610038161003816119905=0
(63)
where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot
120598
Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if
Φ (119909) = P119899119902[119891 (119905)]
1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(64)
Proof We know from [2 17] thatP119899119902[119905]|119905=0
= 119867119899119902 (65)
We thus obtainΦ (119909) = P
119899119902[E119902(119909119905)]
10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]
1003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(66)
Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by
P119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0 (67)
Can be compared with (53)
Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)
Ξ120584119902(119909) andΨ
120584119902(119909) be the 119902-Appell polynomials corresponding
to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then
Ξ120584119902(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119896119902(119911) (68)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)] sdot
120598W119899119902[ℎ (119905)E
119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 oplus119902119911) 119905)]
10038161003816100381610038161003816119905=0
(69)
By (11) the left-hand side of (69) is equal to
(((((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))))
)
times(
Ψ0119902(119911)
Ψ1119902(119911)
Ψ119899minus1119902
(119911)
)
(70)
By (11) the right-hand side of (69) is equal to
(
Ξ0119902(119910 oplus119902119911)
Ξ1119902(119910 oplus119902119911)
Ξ119899minus1119902
(119910 oplus119902119911)
) (71)
The theorem follows by equating the last rows of formulas (71)and (70)
Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which
enables inverses
Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902
and Ξ120584119902
be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902
120584be the pseudo 119902-Appell polynomial
for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then
Ξ120584119902(119910 ⊞119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119902
119896(119911) (72)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)]
10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E
1119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 ⊞119902119911) 119905)] |
119905=0
(73)
Journal of Discrete Mathematics 7
By (11) and (13) the left-hand side of (73) is equal to
(((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))
)
times(
Ψ119902
0(119911)
Ψ119902
1(119911)
Ψ119902
119899minus1(119911)
)
(74)
By (11) the right-hand side of (73) is equal to
(
Ξ0119902(119910 ⊞119902119911)
Ξ1119902(119910 ⊞119902119911)
Ξ119899minus1119902
(119910 ⊞119902119911)
) (75)
The theorem follows by equating the last rows of formulas(75) and (74)
The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)
Corollary 24 ([11 4242]) When sum119904119897=1
119899119897= 119899
B(119899)119873119882119860119896119902
(1199091oplus119902sdot sdot sdot oplus119902119909119904)
= sum
1198981+sdotsdotsdot+119898
119904=119896
(119896
1198981 119898
119904
)
119902
119904
prod
119895=1
B(119899119895)119873119882119860119898
119895119902(119909119895)
(76)
where we assume that 119899119895operates on 119909
119895
Yang and Youn have found a new identity betweenBernoulli and Euler polynomials
2119899B119899(119910 + 119911
2)
120584
sum
119896=0
(120584
119896)B119896(119910)E119899minus119896
(119911) (77)
The 119902-analogue looks as follows
Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider
L119873119882119860120584119902
(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
B119873119882119860120584minus119896119902
(119910) F119873119882119860120584119902
(119911)
(78)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E
119902(119905)+1)
in (68) Then
119892119899(119905) = 119891 (119905) 119892 (119905) =
(2119905)1
(E119902(1199052119902) minus 1)
(79)
This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers
Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)
be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-
mial Then119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(minus119909) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(80)
Let Φ119899119902(119909) and Ψ
119899119902(119909) be 119902-Appell polynomials Then
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910)Ψ119899minus119896119902
(119911) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902
(81)
Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E
119902(119905(119910 oplus
119902119911)) and ℎ(119905)
Or use an equivalent umbral reasoning
Corollary 27 One has the following special cases119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119909)B119902119873119882119860119899minus119896
(minus119909)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
B119873119882119860119899minus119896119902
(82)
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(83)
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(84)
8 Journal of Discrete Mathematics
Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ
119899119902(119909) and the pseudo-119902-Appell polynomial
Ψ119902
119899(119909) Then119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(119909) =
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(85)
Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and
Ψ119902
119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula
(50)
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0
=((
(
Φ0119902(119909) 0 sdot sdot sdot 0
Φ1119902(119909) Φ
0119902(119909) sdot sdot sdot 0
Φ2119902(119909) 2
119902Φ1119902(119909) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909) (119899 minus 1
1)
119902
Φ119899minus2119902
(119909) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909)
))
)
(
Ψ119902
0(119909)
minusΨ119902
1(119909)
(minus1)119899+1
Ψ119902
119899minus1(119909)
)
(86)
On the other hand by formula (49) we obtain
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot
120598W119899119902[ℎ (minus119905)]|
119905=0
=((
(
Φ0119902(0) 0 sdot sdot sdot 0
Φ1119902(0) Φ
0119902(0) sdot sdot sdot 0
Φ2119902(0) 2
119902Φ1119902(0) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(0) (119899 minus 1
1)
119902
Φ119899minus2119902
(0) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(0)
))
)
(
Ψ119902
0(0)
minusΨ119902
1(0)
(minus1)119899+1
Ψ119902
119899minus1(0)
)
(87)
The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and
(87)
We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902 (119909)B119902
NWA119899minus119896 (119909)
=
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902BNWA119899minus119896119902
(88)
Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ
119902(119909) and Ψ
119902(119909) be the 119902-Appell
vectors corresponding to 119891(119905) and 119892(119905) respectively Then
Φ119902(119909) = P
119899119902[119891 (119905)
119892 (119905)]
10038161003816100381610038161003816100381610038161003816119905=0
Ψ119902(119909) (89)
Proof By formula (49) we get
Φ119902(119909) = W
119899119902[119891 (119905)
119892 (119905)
119892 (119905)E119902(119909119905)]
10038161003816100381610038161003816100381610038161003816119905=0
= P119899119902[119891 (119905)
119892 (119905)] sdot120598W119899119902[119892 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(90)
Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials
B119873119882119860119899119902
(119909) =119899119902F119873119882119860119899minus1119902
(119909)
2
+
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119899minus119896119902
F119873119882119860119896119902
(119909)
(91)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E
119902(119905)+1)
in (89) Then
119891 (119905)
119892 (119905)=1199051
2+
1199051
E119902(119905) minus 1
(92)
Journal of Discrete Mathematics 9
We find that
(119891 (119905)
119892 (119905))
(119896)
(0) =
1
2+ BNWA119896119902 if 119896 = 1
BNWA119896119902 otherwise
(93)
The result follows by (89)
Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the
119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies
the following 119902-difference equation
119899
sum
119896=2
119902119899
120572119896minus1
119896 minus 1119902D119896119902119891 (119909119902
minus1
)
+ 119902119899
(119909119902 + 1205720)D119902119891 (119909119902
minus1
)
minus 119899 + 1119902119891 (119909) + 119902
119899
119891 (119909119902minus1
) = 0
(94)
where 120572119896equiv D119896119902(minus119891(119905)D
119902(1119891(119905)))|
119905=0
Proof We put 119891(119905) = 1119892(119905)
(
Φ1119902(119909)
Φ2119902(119909)
Φ119899119902(119909)
) = W119899119902[D119902(E119902(119909119905)
119892 (119905))]
100381610038161003816100381610038161003816100381610038161003816119905=0
= W119899119902[(119909 minus
D119902119892 (119905)
119892 (119905))E119902(119909119905)
119892 (119902119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
by (50)= P
119899119902[E119902(119909119905)
119892 (119902119905)] sdot120598W119899119902[119909 minus
D119902119892 (119905)
119892 (119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
=(((
(
Φ0119902(119909119902minus1
) 0 sdot sdot sdot 0
119902Φ1119902(119909119902minus1
) Φ0119902(119909119902minus1
) sdot sdot sdot 0
1199022
Φ2119902(119909119902minus1
) 1199022119902Φ1119902(119909119902minus1
) sdot sdot sdot 0
119902119899minus1
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909119902minus1
) 119902119899minus2
(119899 minus 1
1)
119902
Φ119899minus2119902
(119909119902minus1
) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909119902minus1
)
)))
)
(
119909+ 1205720
1205721
120572119899minus1
)
(95)
By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)
Φ120584119902(119909) = (119909 + 120572
0) 119902120584minus1
Φ120584minus1119902
(119909119902minus1
)
+
120584minus1
sum
119896=1
(120584 minus 1
119896)
119902
120572119896119902120584minus119896minus1
Φ120584minus119896minus1119902
(119909119902minus1
)
(96)
Operating with D119902on this and putting 120584 minus 1 = 119899
119899 + 1119902Φ119899119902(119909)
= 119902119899
Φ119899119902(119909119902minus1
)
+ 119902119899minus1
119899119902(119902119909 + 120572
0)Φ119899minus1119902
(119909119902minus1
)
+
119899
sum
119896=1
(119899
119896)
119902
120572119896119902119899minus119896minus1
119899 minus 119896119902
times Φ119899minus119896minus1119902
(119909119902minus1
) 119899 = 0 1 119899
(97)
where we have used the following identities
D119896119902Φ119899119902(119909119902minus1
) = 119902minus119896
119899 minus 119896 + 1119896119902Φ119899minus119896119902
D119899119902Φ119899minus1119902
(119909119902minus1
) = 0
(98)
4 Conclusion
By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]
10 Journal of Discrete Mathematics
Appendix
Bernoulli Number Formulas in Hansen
In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows
119899
sum
119896=0
(119899
119896)B119896(119910)B119899minus119896
(119911)
=
119899
sum
119896=0
(119899
119896)B119896(119910 + 119911)B
119899minus119896
(A1)
Hansen [10 page 341 50112] writes this as follows
119899
sum
119896=0
(minus1)119896(minus119899)119896
119896B119896(119910)B119899minus119896
(119911)
= 119899 (119910 + 119911 minus 1)B119899minus1
(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)
(A2)
Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials
References
[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012
[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted
[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963
[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923
[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009
[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967
[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006
[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007
[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009
[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser
2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169
Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981
[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951
[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition
[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009
[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted
[18] N E Noslashrlund Differenzenrechnung Springer 1924
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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4 Journal of Discrete Mathematics
(119903 + 119904
119895)
119902
Ψ1015840
119903+119904minus119895119902
= 119891 (119906) 119892 (119907)
infin
sum
119896=119895
(119896
119895)
119902
Ψ1015840
119896minus119895119902sum
119903+119904=119896
(119886119906)119903
(119887119907)119904
119903119902119904119902
= 119891 (119906) 119892 (119907)
infin
sum
119896=119895
(119896
119895)
119902
Ψ1015840
119896minus119895119902
(119886119906 oplus119902119887119907)119896
119896119902
(26)
23 Characterization of 119902-Appell Numbers We will now pre-sent a characterization of 119902-Appell numbers comparewithAl-Salam [6]
Definition 7 We denote by A119902the set of all 119902-Appell-
numbersLet Φ
119899119902and Ψ
119899119902be two elements in A
119902 Then the
operations + and ⋆ are defined as follows
(Φ119902+ Ψ119902)119899
equiv Φ119899119902+ Ψ119899119902 (27)
(Φ119902⋆ Ψ119902)119899
equiv
119899
sum
119896=0
(119899
119896)
119902
Φ119899minus119896119902
Ψ119896119902 (28)
Theorem 8 The identity element in A119902with respect to ⋆ is
119868 equiv 1205751198990 This corresponds to 119891(119905) = 1
Proof Put 119896 = 119899 and Φ119902= 1 in (28)
Theorem 9 (Compare with [6 page 34]) Let Φ119899119902 Ξ119899119902
andΨ119899119902
be three elements in A119902corresponding to the generating
functions119891(119905) 119892(119905) and ℎ(119905) respectively Further assume that119891(0) + ℎ(0) = 0 Then
Φ119899119902+ Ψ119899119902
isin A119902
Φ119899119902+ Ψ119899119902
has generating function119891 (119905) + ℎ (119905) (29)
The associative law for addition reads
Φ119899119902+ (Ξ119899119902+ Ψ119899119902) = (Φ
119899119902+ Ξ119899119902) + Ψ119899119902 (30)
Proof This follows from the definitions
Theorem 10 (Compare with [6 page 34]) LetΦ119899119902 Ξ119899119902 and
Ψ119899119902
be three elements in A119902corresponding to the generating
functions 119891(119905) 119892(119905) and ℎ(119905) respectively Then
Φ119899119902⋆ Ψ119899119902
isin A119902 (31)
Φ119899119902⋆ Ψ119899119902
= Ψ119899119902⋆ Φ119899119902 (32)
Φ119899119902⋆ Ψ119899119902
has generating function119891 (119905) ℎ (119905) (33)
The associative law for ⋆ reads
Φ119899119902⋆ (Ξ119899119902⋆ Ψ119899119902) = (Φ
119899119902⋆ Ξ119899119902) ⋆ Ψ119899119902 (34)
Proof We prove (31) and (33) at the same time We have inthe umbral sense
119891 (119905) ℎ (119905) = E119902(119905 (Φ119902oplus119902Ψ119902)) (35)
Formulas (32) and (34) follow by the commutativity andthe associativity of NWA
24 Specializations
Definition 11 The NWA 119902-Bernoulli polynomialsBNWA120584119902(119909) of degree 120584 are defined by
1199051
(E119902(119905) minus 1)
E119902(119909119905) =
infin
sum
120584=0
119905120584BNWA120584119902 (119909)
120584119902
|119905| lt 2120587 (36)
The NWA 119902-Bernoulli polynomials are also given by
BNWA120584119902 (119909) = (BNWA119902 oplus119902 119909)120584
(37)
The pseudo-119902-Bernoulli polynomials are defined by
B119902NWA120584 (119909) = (BNWA119902 ⊞119902 119909)120584
(38)
The JHC 119902-Bernoulli polynomials BJHC120584119902(119909) of degree 120584are defined by
1199051
(E1119902
(119905) minus 1)E119902(119909119905) =
infin
sum
120584=0
119905120584BJHC120584119902 (119909)
120584119902
|119905| lt 2120587 (39)
3 119902-Pascal and 119902-Wronskian Matrices
We now come to the most important part of the paper whichis of interest even for the case 119902 = 1 since the original paperby Yang and Youn is not very widespread
Definition 12 The generalized 119902-Pascal functional matrix isgiven by
(P119899119902) [119891 (119905 119902)] (119894 119895)
equiv
(119894
119895)
119902
D119894minus119895119902119905119891 (119905 119902) if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(40)
In the special case 119891(119905 119902) = constant
(P119899119902) [119891 (119905 119902)] = 119891 (119905 119902) 119868 (41)
Definition 13 The 119902-deformed Wronskian vector is given by
(W119899119902) [119891 (119905 119902)] (119894 0) equiv D119894
119902119905119891 (119905 119902) 119894 = 0 1 119899 minus 1
(42)
Journal of Discrete Mathematics 5
Definition 14 The 119902-deformed Leibniz functional matrix isgiven by(L119899119902) [119891 (119905 119902)] (119894 119895)
equiv
D119894minus119895119902119905119891 (119905 119902)
119894 minus 119895119902
if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(43)
In order to find a 119902-analogue of the formula [16]
(L119899) [119891 (119905) 119892 (119905)] = (L
119899) [119891 (119905)] (L
119899) [119892 (119905)] (44)
we infer that by the 119902-Leibniz formula [1](L119899119902) [119891 (119905 119902) 119892 (119905 119902)]
= (L119899119902) [119891 (119905 119902)] sdot
120598(L119899119902) [119892 (119905 119902)]
(45)
where in the matrix multiplication for every term whichincludes D119896
119902119891 we operate with 120598119896 on 119892 We denote this by sdot
120598
This operator can also be iterated compare with [1]
Theorem 15 (a 119902-analogue of [9 page 67]) Consider
P119899119902[E119902(119909119905)]
10038161003816100381610038161003816119905=0= 119875119899119902(119909) (46)
where the 119902-Pascal matrix is defined in (8)The mappingsP
119899119902[sdot] andW
119899119902[sdot] are linear that is
P119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886P
119899119902[119891 (119905)] + 119887P
119899119902[119892 (119905)]
W119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886W
119899119902[119891 (119905)] + 119887W
119899119902[119892 (119905)]
(47)
One has the following two matrix multiplication formulasthe first one is a 119902-analogue of [8 page 232]
P119899119902[119891 (119905 119902)] sdot
120598P119899119902[119892 (119905 119902)] = P
119899119902[119891 (119905 119902) 119892 (119905 119902)]
(48)
P119899119902[119891 (119905 119902)] sdot
120598W119899119902[119892 (119905 119902)] = W
119899119902[119891 (119905 119902) 119892 (119905 119902)]
(49)
The scaling properties can be stated as follows
W119899119902[119891 (119886119905 119902)]
1003816100381610038161003816119905=0
= diag [1 119886 119886119899minus1]W119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0
(50)
P119899119902[119891 (119886119905 119902)]
1003816100381610038161003816119905=0
= diag [1 119886 119886119899minus1]P119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0
times diag [1 119886minus1 1198861minus119899]
(51)
Proof The formula (49) is proved by the 119902-Leibniz theoremFor (50) use the chain rule
Definition 16 Assume that 119891(0) = 0 If the inverse (119891(119905)minus1)(119896)exists for 119896 lt 119899 we can define the 119902-inverse of the generalized119902-Pascal functional matrix as
P119899119902[119891 (119905 119902)]
minus1119902
equiv P119899119902[119891(119905 119902)
minus1
] (52)
Example 17 (a 119902-analogue of [8 pages 231ndash234]) Let
(G119899119902) [119909] (119894 119895)
equiv
119892119894minus119895(119909) (
119894
119895)
119902
if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(53)
Put
119891 (119905 119909 119902) equiv
119899
sum
119896=0
119892119896(119909)
119905119896
119896119902 (54)
a truncated 119902-exponential generating function of 119902-binomialtype We find that
P119899119902[119891 (119905 119909 119902)]
1003816100381610038161003816119905=0 = G119899119902[119909] (55)
By a routine computation we obtain
G119899119902[119909]G119899119902[119910] = G
119899119902[119909 oplus119902119910] (56)
The function argument 119909oplus119902119910 is induced from the corre-
sponding 119902-additionIn the same way we obtain
G119899119902[119909]G1198991119902
[119910] = G119899119902[119909 ⊞119902119910] (57)
By the last formula we can define the inverse ofG119899119902[119909] as
G119899119902[119909]minus1
equiv G1198991119902
[minus119909] (58)
We now return permanently to [9]
Definition 18 (compare with [9 page 71]) LetΦ119902(119909) be the 119902-
Appell polynomial corresponding to 119891(119905) in (11) The vectorof 119902-Appell polynomials Φ
119902(119909) for 119891(119905) is given by
Φ119902(119909) (119894) equiv Φ
119894119902(119909) 119894 = 0 1 119899 minus 1 (59)
Theorem 19 (a 119902-analogue of [9 page 69]) Let 119891(119905) beanalytic in the neighbourhood of 0 with 119891(0) = 0 The 119902-difference equation in R119899
(D119902119909) 119910 (119909) = P
119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0 119910 (119909) minusinfin lt 119909 lt infin
(60)
with initial value 119910(0) = 1199100 has the following solution
119910 (119909) = P119899119902[E119902(119891 (119905) 119909)]
10038161003816100381610038161003816119905=01199100 (61)
Proof Since E119902(119860119905)|119905=0
is equal to the unit matrix the solu-tion of (60) is
Φ119902(119909) = E
119902(P119899119902[119891 (119905)]
1003816100381610038161003816119905=0119909) 1199100
= (
infin
sum
119896=0
P119896
119899119902[119891 (119905)])
1003816100381610038161003816100381610038161003816100381610038161003816119905=0
119909119896
119896119902
(62)
6 Journal of Discrete Mathematics
By (47) and (48)
Φ119902(119909) = P
119899119902(
infin
sum
119896=0
119891119896
(119905)119909119896
119896119902)
1003816100381610038161003816100381610038161003816100381610038161003816119905=0
(63)
where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot
120598
Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if
Φ (119909) = P119899119902[119891 (119905)]
1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(64)
Proof We know from [2 17] thatP119899119902[119905]|119905=0
= 119867119899119902 (65)
We thus obtainΦ (119909) = P
119899119902[E119902(119909119905)]
10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]
1003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(66)
Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by
P119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0 (67)
Can be compared with (53)
Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)
Ξ120584119902(119909) andΨ
120584119902(119909) be the 119902-Appell polynomials corresponding
to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then
Ξ120584119902(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119896119902(119911) (68)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)] sdot
120598W119899119902[ℎ (119905)E
119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 oplus119902119911) 119905)]
10038161003816100381610038161003816119905=0
(69)
By (11) the left-hand side of (69) is equal to
(((((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))))
)
times(
Ψ0119902(119911)
Ψ1119902(119911)
Ψ119899minus1119902
(119911)
)
(70)
By (11) the right-hand side of (69) is equal to
(
Ξ0119902(119910 oplus119902119911)
Ξ1119902(119910 oplus119902119911)
Ξ119899minus1119902
(119910 oplus119902119911)
) (71)
The theorem follows by equating the last rows of formulas (71)and (70)
Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which
enables inverses
Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902
and Ξ120584119902
be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902
120584be the pseudo 119902-Appell polynomial
for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then
Ξ120584119902(119910 ⊞119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119902
119896(119911) (72)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)]
10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E
1119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 ⊞119902119911) 119905)] |
119905=0
(73)
Journal of Discrete Mathematics 7
By (11) and (13) the left-hand side of (73) is equal to
(((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))
)
times(
Ψ119902
0(119911)
Ψ119902
1(119911)
Ψ119902
119899minus1(119911)
)
(74)
By (11) the right-hand side of (73) is equal to
(
Ξ0119902(119910 ⊞119902119911)
Ξ1119902(119910 ⊞119902119911)
Ξ119899minus1119902
(119910 ⊞119902119911)
) (75)
The theorem follows by equating the last rows of formulas(75) and (74)
The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)
Corollary 24 ([11 4242]) When sum119904119897=1
119899119897= 119899
B(119899)119873119882119860119896119902
(1199091oplus119902sdot sdot sdot oplus119902119909119904)
= sum
1198981+sdotsdotsdot+119898
119904=119896
(119896
1198981 119898
119904
)
119902
119904
prod
119895=1
B(119899119895)119873119882119860119898
119895119902(119909119895)
(76)
where we assume that 119899119895operates on 119909
119895
Yang and Youn have found a new identity betweenBernoulli and Euler polynomials
2119899B119899(119910 + 119911
2)
120584
sum
119896=0
(120584
119896)B119896(119910)E119899minus119896
(119911) (77)
The 119902-analogue looks as follows
Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider
L119873119882119860120584119902
(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
B119873119882119860120584minus119896119902
(119910) F119873119882119860120584119902
(119911)
(78)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E
119902(119905)+1)
in (68) Then
119892119899(119905) = 119891 (119905) 119892 (119905) =
(2119905)1
(E119902(1199052119902) minus 1)
(79)
This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers
Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)
be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-
mial Then119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(minus119909) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(80)
Let Φ119899119902(119909) and Ψ
119899119902(119909) be 119902-Appell polynomials Then
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910)Ψ119899minus119896119902
(119911) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902
(81)
Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E
119902(119905(119910 oplus
119902119911)) and ℎ(119905)
Or use an equivalent umbral reasoning
Corollary 27 One has the following special cases119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119909)B119902119873119882119860119899minus119896
(minus119909)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
B119873119882119860119899minus119896119902
(82)
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(83)
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(84)
8 Journal of Discrete Mathematics
Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ
119899119902(119909) and the pseudo-119902-Appell polynomial
Ψ119902
119899(119909) Then119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(119909) =
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(85)
Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and
Ψ119902
119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula
(50)
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0
=((
(
Φ0119902(119909) 0 sdot sdot sdot 0
Φ1119902(119909) Φ
0119902(119909) sdot sdot sdot 0
Φ2119902(119909) 2
119902Φ1119902(119909) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909) (119899 minus 1
1)
119902
Φ119899minus2119902
(119909) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909)
))
)
(
Ψ119902
0(119909)
minusΨ119902
1(119909)
(minus1)119899+1
Ψ119902
119899minus1(119909)
)
(86)
On the other hand by formula (49) we obtain
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot
120598W119899119902[ℎ (minus119905)]|
119905=0
=((
(
Φ0119902(0) 0 sdot sdot sdot 0
Φ1119902(0) Φ
0119902(0) sdot sdot sdot 0
Φ2119902(0) 2
119902Φ1119902(0) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(0) (119899 minus 1
1)
119902
Φ119899minus2119902
(0) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(0)
))
)
(
Ψ119902
0(0)
minusΨ119902
1(0)
(minus1)119899+1
Ψ119902
119899minus1(0)
)
(87)
The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and
(87)
We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902 (119909)B119902
NWA119899minus119896 (119909)
=
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902BNWA119899minus119896119902
(88)
Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ
119902(119909) and Ψ
119902(119909) be the 119902-Appell
vectors corresponding to 119891(119905) and 119892(119905) respectively Then
Φ119902(119909) = P
119899119902[119891 (119905)
119892 (119905)]
10038161003816100381610038161003816100381610038161003816119905=0
Ψ119902(119909) (89)
Proof By formula (49) we get
Φ119902(119909) = W
119899119902[119891 (119905)
119892 (119905)
119892 (119905)E119902(119909119905)]
10038161003816100381610038161003816100381610038161003816119905=0
= P119899119902[119891 (119905)
119892 (119905)] sdot120598W119899119902[119892 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(90)
Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials
B119873119882119860119899119902
(119909) =119899119902F119873119882119860119899minus1119902
(119909)
2
+
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119899minus119896119902
F119873119882119860119896119902
(119909)
(91)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E
119902(119905)+1)
in (89) Then
119891 (119905)
119892 (119905)=1199051
2+
1199051
E119902(119905) minus 1
(92)
Journal of Discrete Mathematics 9
We find that
(119891 (119905)
119892 (119905))
(119896)
(0) =
1
2+ BNWA119896119902 if 119896 = 1
BNWA119896119902 otherwise
(93)
The result follows by (89)
Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the
119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies
the following 119902-difference equation
119899
sum
119896=2
119902119899
120572119896minus1
119896 minus 1119902D119896119902119891 (119909119902
minus1
)
+ 119902119899
(119909119902 + 1205720)D119902119891 (119909119902
minus1
)
minus 119899 + 1119902119891 (119909) + 119902
119899
119891 (119909119902minus1
) = 0
(94)
where 120572119896equiv D119896119902(minus119891(119905)D
119902(1119891(119905)))|
119905=0
Proof We put 119891(119905) = 1119892(119905)
(
Φ1119902(119909)
Φ2119902(119909)
Φ119899119902(119909)
) = W119899119902[D119902(E119902(119909119905)
119892 (119905))]
100381610038161003816100381610038161003816100381610038161003816119905=0
= W119899119902[(119909 minus
D119902119892 (119905)
119892 (119905))E119902(119909119905)
119892 (119902119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
by (50)= P
119899119902[E119902(119909119905)
119892 (119902119905)] sdot120598W119899119902[119909 minus
D119902119892 (119905)
119892 (119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
=(((
(
Φ0119902(119909119902minus1
) 0 sdot sdot sdot 0
119902Φ1119902(119909119902minus1
) Φ0119902(119909119902minus1
) sdot sdot sdot 0
1199022
Φ2119902(119909119902minus1
) 1199022119902Φ1119902(119909119902minus1
) sdot sdot sdot 0
119902119899minus1
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909119902minus1
) 119902119899minus2
(119899 minus 1
1)
119902
Φ119899minus2119902
(119909119902minus1
) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909119902minus1
)
)))
)
(
119909+ 1205720
1205721
120572119899minus1
)
(95)
By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)
Φ120584119902(119909) = (119909 + 120572
0) 119902120584minus1
Φ120584minus1119902
(119909119902minus1
)
+
120584minus1
sum
119896=1
(120584 minus 1
119896)
119902
120572119896119902120584minus119896minus1
Φ120584minus119896minus1119902
(119909119902minus1
)
(96)
Operating with D119902on this and putting 120584 minus 1 = 119899
119899 + 1119902Φ119899119902(119909)
= 119902119899
Φ119899119902(119909119902minus1
)
+ 119902119899minus1
119899119902(119902119909 + 120572
0)Φ119899minus1119902
(119909119902minus1
)
+
119899
sum
119896=1
(119899
119896)
119902
120572119896119902119899minus119896minus1
119899 minus 119896119902
times Φ119899minus119896minus1119902
(119909119902minus1
) 119899 = 0 1 119899
(97)
where we have used the following identities
D119896119902Φ119899119902(119909119902minus1
) = 119902minus119896
119899 minus 119896 + 1119896119902Φ119899minus119896119902
D119899119902Φ119899minus1119902
(119909119902minus1
) = 0
(98)
4 Conclusion
By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]
10 Journal of Discrete Mathematics
Appendix
Bernoulli Number Formulas in Hansen
In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows
119899
sum
119896=0
(119899
119896)B119896(119910)B119899minus119896
(119911)
=
119899
sum
119896=0
(119899
119896)B119896(119910 + 119911)B
119899minus119896
(A1)
Hansen [10 page 341 50112] writes this as follows
119899
sum
119896=0
(minus1)119896(minus119899)119896
119896B119896(119910)B119899minus119896
(119911)
= 119899 (119910 + 119911 minus 1)B119899minus1
(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)
(A2)
Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials
References
[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012
[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted
[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963
[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923
[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009
[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967
[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006
[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007
[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009
[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser
2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169
Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981
[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951
[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition
[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009
[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted
[18] N E Noslashrlund Differenzenrechnung Springer 1924
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Discrete Mathematics 5
Definition 14 The 119902-deformed Leibniz functional matrix isgiven by(L119899119902) [119891 (119905 119902)] (119894 119895)
equiv
D119894minus119895119902119905119891 (119905 119902)
119894 minus 119895119902
if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(43)
In order to find a 119902-analogue of the formula [16]
(L119899) [119891 (119905) 119892 (119905)] = (L
119899) [119891 (119905)] (L
119899) [119892 (119905)] (44)
we infer that by the 119902-Leibniz formula [1](L119899119902) [119891 (119905 119902) 119892 (119905 119902)]
= (L119899119902) [119891 (119905 119902)] sdot
120598(L119899119902) [119892 (119905 119902)]
(45)
where in the matrix multiplication for every term whichincludes D119896
119902119891 we operate with 120598119896 on 119892 We denote this by sdot
120598
This operator can also be iterated compare with [1]
Theorem 15 (a 119902-analogue of [9 page 67]) Consider
P119899119902[E119902(119909119905)]
10038161003816100381610038161003816119905=0= 119875119899119902(119909) (46)
where the 119902-Pascal matrix is defined in (8)The mappingsP
119899119902[sdot] andW
119899119902[sdot] are linear that is
P119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886P
119899119902[119891 (119905)] + 119887P
119899119902[119892 (119905)]
W119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886W
119899119902[119891 (119905)] + 119887W
119899119902[119892 (119905)]
(47)
One has the following two matrix multiplication formulasthe first one is a 119902-analogue of [8 page 232]
P119899119902[119891 (119905 119902)] sdot
120598P119899119902[119892 (119905 119902)] = P
119899119902[119891 (119905 119902) 119892 (119905 119902)]
(48)
P119899119902[119891 (119905 119902)] sdot
120598W119899119902[119892 (119905 119902)] = W
119899119902[119891 (119905 119902) 119892 (119905 119902)]
(49)
The scaling properties can be stated as follows
W119899119902[119891 (119886119905 119902)]
1003816100381610038161003816119905=0
= diag [1 119886 119886119899minus1]W119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0
(50)
P119899119902[119891 (119886119905 119902)]
1003816100381610038161003816119905=0
= diag [1 119886 119886119899minus1]P119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0
times diag [1 119886minus1 1198861minus119899]
(51)
Proof The formula (49) is proved by the 119902-Leibniz theoremFor (50) use the chain rule
Definition 16 Assume that 119891(0) = 0 If the inverse (119891(119905)minus1)(119896)exists for 119896 lt 119899 we can define the 119902-inverse of the generalized119902-Pascal functional matrix as
P119899119902[119891 (119905 119902)]
minus1119902
equiv P119899119902[119891(119905 119902)
minus1
] (52)
Example 17 (a 119902-analogue of [8 pages 231ndash234]) Let
(G119899119902) [119909] (119894 119895)
equiv
119892119894minus119895(119909) (
119894
119895)
119902
if 119894 ge 119895
119894 119895 = 0 1 2 119899 minus 1
0 otherwise(53)
Put
119891 (119905 119909 119902) equiv
119899
sum
119896=0
119892119896(119909)
119905119896
119896119902 (54)
a truncated 119902-exponential generating function of 119902-binomialtype We find that
P119899119902[119891 (119905 119909 119902)]
1003816100381610038161003816119905=0 = G119899119902[119909] (55)
By a routine computation we obtain
G119899119902[119909]G119899119902[119910] = G
119899119902[119909 oplus119902119910] (56)
The function argument 119909oplus119902119910 is induced from the corre-
sponding 119902-additionIn the same way we obtain
G119899119902[119909]G1198991119902
[119910] = G119899119902[119909 ⊞119902119910] (57)
By the last formula we can define the inverse ofG119899119902[119909] as
G119899119902[119909]minus1
equiv G1198991119902
[minus119909] (58)
We now return permanently to [9]
Definition 18 (compare with [9 page 71]) LetΦ119902(119909) be the 119902-
Appell polynomial corresponding to 119891(119905) in (11) The vectorof 119902-Appell polynomials Φ
119902(119909) for 119891(119905) is given by
Φ119902(119909) (119894) equiv Φ
119894119902(119909) 119894 = 0 1 119899 minus 1 (59)
Theorem 19 (a 119902-analogue of [9 page 69]) Let 119891(119905) beanalytic in the neighbourhood of 0 with 119891(0) = 0 The 119902-difference equation in R119899
(D119902119909) 119910 (119909) = P
119899119902[119891 (119905 119902)]
1003816100381610038161003816119905=0 119910 (119909) minusinfin lt 119909 lt infin
(60)
with initial value 119910(0) = 1199100 has the following solution
119910 (119909) = P119899119902[E119902(119891 (119905) 119909)]
10038161003816100381610038161003816119905=01199100 (61)
Proof Since E119902(119860119905)|119905=0
is equal to the unit matrix the solu-tion of (60) is
Φ119902(119909) = E
119902(P119899119902[119891 (119905)]
1003816100381610038161003816119905=0119909) 1199100
= (
infin
sum
119896=0
P119896
119899119902[119891 (119905)])
1003816100381610038161003816100381610038161003816100381610038161003816119905=0
119909119896
119896119902
(62)
6 Journal of Discrete Mathematics
By (47) and (48)
Φ119902(119909) = P
119899119902(
infin
sum
119896=0
119891119896
(119905)119909119896
119896119902)
1003816100381610038161003816100381610038161003816100381610038161003816119905=0
(63)
where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot
120598
Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if
Φ (119909) = P119899119902[119891 (119905)]
1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(64)
Proof We know from [2 17] thatP119899119902[119905]|119905=0
= 119867119899119902 (65)
We thus obtainΦ (119909) = P
119899119902[E119902(119909119905)]
10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]
1003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(66)
Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by
P119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0 (67)
Can be compared with (53)
Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)
Ξ120584119902(119909) andΨ
120584119902(119909) be the 119902-Appell polynomials corresponding
to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then
Ξ120584119902(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119896119902(119911) (68)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)] sdot
120598W119899119902[ℎ (119905)E
119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 oplus119902119911) 119905)]
10038161003816100381610038161003816119905=0
(69)
By (11) the left-hand side of (69) is equal to
(((((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))))
)
times(
Ψ0119902(119911)
Ψ1119902(119911)
Ψ119899minus1119902
(119911)
)
(70)
By (11) the right-hand side of (69) is equal to
(
Ξ0119902(119910 oplus119902119911)
Ξ1119902(119910 oplus119902119911)
Ξ119899minus1119902
(119910 oplus119902119911)
) (71)
The theorem follows by equating the last rows of formulas (71)and (70)
Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which
enables inverses
Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902
and Ξ120584119902
be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902
120584be the pseudo 119902-Appell polynomial
for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then
Ξ120584119902(119910 ⊞119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119902
119896(119911) (72)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)]
10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E
1119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 ⊞119902119911) 119905)] |
119905=0
(73)
Journal of Discrete Mathematics 7
By (11) and (13) the left-hand side of (73) is equal to
(((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))
)
times(
Ψ119902
0(119911)
Ψ119902
1(119911)
Ψ119902
119899minus1(119911)
)
(74)
By (11) the right-hand side of (73) is equal to
(
Ξ0119902(119910 ⊞119902119911)
Ξ1119902(119910 ⊞119902119911)
Ξ119899minus1119902
(119910 ⊞119902119911)
) (75)
The theorem follows by equating the last rows of formulas(75) and (74)
The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)
Corollary 24 ([11 4242]) When sum119904119897=1
119899119897= 119899
B(119899)119873119882119860119896119902
(1199091oplus119902sdot sdot sdot oplus119902119909119904)
= sum
1198981+sdotsdotsdot+119898
119904=119896
(119896
1198981 119898
119904
)
119902
119904
prod
119895=1
B(119899119895)119873119882119860119898
119895119902(119909119895)
(76)
where we assume that 119899119895operates on 119909
119895
Yang and Youn have found a new identity betweenBernoulli and Euler polynomials
2119899B119899(119910 + 119911
2)
120584
sum
119896=0
(120584
119896)B119896(119910)E119899minus119896
(119911) (77)
The 119902-analogue looks as follows
Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider
L119873119882119860120584119902
(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
B119873119882119860120584minus119896119902
(119910) F119873119882119860120584119902
(119911)
(78)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E
119902(119905)+1)
in (68) Then
119892119899(119905) = 119891 (119905) 119892 (119905) =
(2119905)1
(E119902(1199052119902) minus 1)
(79)
This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers
Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)
be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-
mial Then119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(minus119909) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(80)
Let Φ119899119902(119909) and Ψ
119899119902(119909) be 119902-Appell polynomials Then
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910)Ψ119899minus119896119902
(119911) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902
(81)
Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E
119902(119905(119910 oplus
119902119911)) and ℎ(119905)
Or use an equivalent umbral reasoning
Corollary 27 One has the following special cases119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119909)B119902119873119882119860119899minus119896
(minus119909)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
B119873119882119860119899minus119896119902
(82)
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(83)
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(84)
8 Journal of Discrete Mathematics
Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ
119899119902(119909) and the pseudo-119902-Appell polynomial
Ψ119902
119899(119909) Then119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(119909) =
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(85)
Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and
Ψ119902
119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula
(50)
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0
=((
(
Φ0119902(119909) 0 sdot sdot sdot 0
Φ1119902(119909) Φ
0119902(119909) sdot sdot sdot 0
Φ2119902(119909) 2
119902Φ1119902(119909) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909) (119899 minus 1
1)
119902
Φ119899minus2119902
(119909) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909)
))
)
(
Ψ119902
0(119909)
minusΨ119902
1(119909)
(minus1)119899+1
Ψ119902
119899minus1(119909)
)
(86)
On the other hand by formula (49) we obtain
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot
120598W119899119902[ℎ (minus119905)]|
119905=0
=((
(
Φ0119902(0) 0 sdot sdot sdot 0
Φ1119902(0) Φ
0119902(0) sdot sdot sdot 0
Φ2119902(0) 2
119902Φ1119902(0) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(0) (119899 minus 1
1)
119902
Φ119899minus2119902
(0) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(0)
))
)
(
Ψ119902
0(0)
minusΨ119902
1(0)
(minus1)119899+1
Ψ119902
119899minus1(0)
)
(87)
The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and
(87)
We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902 (119909)B119902
NWA119899minus119896 (119909)
=
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902BNWA119899minus119896119902
(88)
Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ
119902(119909) and Ψ
119902(119909) be the 119902-Appell
vectors corresponding to 119891(119905) and 119892(119905) respectively Then
Φ119902(119909) = P
119899119902[119891 (119905)
119892 (119905)]
10038161003816100381610038161003816100381610038161003816119905=0
Ψ119902(119909) (89)
Proof By formula (49) we get
Φ119902(119909) = W
119899119902[119891 (119905)
119892 (119905)
119892 (119905)E119902(119909119905)]
10038161003816100381610038161003816100381610038161003816119905=0
= P119899119902[119891 (119905)
119892 (119905)] sdot120598W119899119902[119892 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(90)
Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials
B119873119882119860119899119902
(119909) =119899119902F119873119882119860119899minus1119902
(119909)
2
+
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119899minus119896119902
F119873119882119860119896119902
(119909)
(91)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E
119902(119905)+1)
in (89) Then
119891 (119905)
119892 (119905)=1199051
2+
1199051
E119902(119905) minus 1
(92)
Journal of Discrete Mathematics 9
We find that
(119891 (119905)
119892 (119905))
(119896)
(0) =
1
2+ BNWA119896119902 if 119896 = 1
BNWA119896119902 otherwise
(93)
The result follows by (89)
Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the
119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies
the following 119902-difference equation
119899
sum
119896=2
119902119899
120572119896minus1
119896 minus 1119902D119896119902119891 (119909119902
minus1
)
+ 119902119899
(119909119902 + 1205720)D119902119891 (119909119902
minus1
)
minus 119899 + 1119902119891 (119909) + 119902
119899
119891 (119909119902minus1
) = 0
(94)
where 120572119896equiv D119896119902(minus119891(119905)D
119902(1119891(119905)))|
119905=0
Proof We put 119891(119905) = 1119892(119905)
(
Φ1119902(119909)
Φ2119902(119909)
Φ119899119902(119909)
) = W119899119902[D119902(E119902(119909119905)
119892 (119905))]
100381610038161003816100381610038161003816100381610038161003816119905=0
= W119899119902[(119909 minus
D119902119892 (119905)
119892 (119905))E119902(119909119905)
119892 (119902119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
by (50)= P
119899119902[E119902(119909119905)
119892 (119902119905)] sdot120598W119899119902[119909 minus
D119902119892 (119905)
119892 (119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
=(((
(
Φ0119902(119909119902minus1
) 0 sdot sdot sdot 0
119902Φ1119902(119909119902minus1
) Φ0119902(119909119902minus1
) sdot sdot sdot 0
1199022
Φ2119902(119909119902minus1
) 1199022119902Φ1119902(119909119902minus1
) sdot sdot sdot 0
119902119899minus1
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909119902minus1
) 119902119899minus2
(119899 minus 1
1)
119902
Φ119899minus2119902
(119909119902minus1
) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909119902minus1
)
)))
)
(
119909+ 1205720
1205721
120572119899minus1
)
(95)
By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)
Φ120584119902(119909) = (119909 + 120572
0) 119902120584minus1
Φ120584minus1119902
(119909119902minus1
)
+
120584minus1
sum
119896=1
(120584 minus 1
119896)
119902
120572119896119902120584minus119896minus1
Φ120584minus119896minus1119902
(119909119902minus1
)
(96)
Operating with D119902on this and putting 120584 minus 1 = 119899
119899 + 1119902Φ119899119902(119909)
= 119902119899
Φ119899119902(119909119902minus1
)
+ 119902119899minus1
119899119902(119902119909 + 120572
0)Φ119899minus1119902
(119909119902minus1
)
+
119899
sum
119896=1
(119899
119896)
119902
120572119896119902119899minus119896minus1
119899 minus 119896119902
times Φ119899minus119896minus1119902
(119909119902minus1
) 119899 = 0 1 119899
(97)
where we have used the following identities
D119896119902Φ119899119902(119909119902minus1
) = 119902minus119896
119899 minus 119896 + 1119896119902Φ119899minus119896119902
D119899119902Φ119899minus1119902
(119909119902minus1
) = 0
(98)
4 Conclusion
By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]
10 Journal of Discrete Mathematics
Appendix
Bernoulli Number Formulas in Hansen
In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows
119899
sum
119896=0
(119899
119896)B119896(119910)B119899minus119896
(119911)
=
119899
sum
119896=0
(119899
119896)B119896(119910 + 119911)B
119899minus119896
(A1)
Hansen [10 page 341 50112] writes this as follows
119899
sum
119896=0
(minus1)119896(minus119899)119896
119896B119896(119910)B119899minus119896
(119911)
= 119899 (119910 + 119911 minus 1)B119899minus1
(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)
(A2)
Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials
References
[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012
[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted
[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963
[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923
[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009
[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967
[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006
[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007
[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009
[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser
2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169
Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981
[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951
[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition
[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009
[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted
[18] N E Noslashrlund Differenzenrechnung Springer 1924
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Discrete Mathematics
By (47) and (48)
Φ119902(119909) = P
119899119902(
infin
sum
119896=0
119891119896
(119905)119909119896
119896119902)
1003816100381610038161003816100381610038161003816100381610038161003816119905=0
(63)
where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot
120598
Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if
Φ (119909) = P119899119902[119891 (119905)]
1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(64)
Proof We know from [2 17] thatP119899119902[119905]|119905=0
= 119867119899119902 (65)
We thus obtainΦ (119909) = P
119899119902[E119902(119909119905)]
10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]
1003816100381610038161003816119905=0
= W119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(66)
Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by
P119899119902[119891 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0 (67)
Can be compared with (53)
Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)
Ξ120584119902(119909) andΨ
120584119902(119909) be the 119902-Appell polynomials corresponding
to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then
Ξ120584119902(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119896119902(119911) (68)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)] sdot
120598W119899119902[ℎ (119905)E
119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 oplus119902119911) 119905)]
10038161003816100381610038161003816119905=0
(69)
By (11) the left-hand side of (69) is equal to
(((((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))))
)
times(
Ψ0119902(119911)
Ψ1119902(119911)
Ψ119899minus1119902
(119911)
)
(70)
By (11) the right-hand side of (69) is equal to
(
Ξ0119902(119910 oplus119902119911)
Ξ1119902(119910 oplus119902119911)
Ξ119899minus1119902
(119910 oplus119902119911)
) (71)
The theorem follows by equating the last rows of formulas (71)and (70)
Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which
enables inverses
Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902
and Ξ120584119902
be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902
120584be the pseudo 119902-Appell polynomial
for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then
Ξ120584119902(119910 ⊞119902119911) =
120584
sum
119896=0
(120584
119896)
119902
Φ120584minus119896119902
(119910)Ψ119902
119896(119911) (72)
Proof By formula (49) we get
P119899119902[119891 (119905)E
119902(119910119905)]
10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E
1119902(119911119905)]
10038161003816100381610038161003816119905=0
= W119899119902[119891 (119905) ℎ (119905)E
119902((119910 ⊞119902119911) 119905)] |
119905=0
(73)
Journal of Discrete Mathematics 7
By (11) and (13) the left-hand side of (73) is equal to
(((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))
)
times(
Ψ119902
0(119911)
Ψ119902
1(119911)
Ψ119902
119899minus1(119911)
)
(74)
By (11) the right-hand side of (73) is equal to
(
Ξ0119902(119910 ⊞119902119911)
Ξ1119902(119910 ⊞119902119911)
Ξ119899minus1119902
(119910 ⊞119902119911)
) (75)
The theorem follows by equating the last rows of formulas(75) and (74)
The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)
Corollary 24 ([11 4242]) When sum119904119897=1
119899119897= 119899
B(119899)119873119882119860119896119902
(1199091oplus119902sdot sdot sdot oplus119902119909119904)
= sum
1198981+sdotsdotsdot+119898
119904=119896
(119896
1198981 119898
119904
)
119902
119904
prod
119895=1
B(119899119895)119873119882119860119898
119895119902(119909119895)
(76)
where we assume that 119899119895operates on 119909
119895
Yang and Youn have found a new identity betweenBernoulli and Euler polynomials
2119899B119899(119910 + 119911
2)
120584
sum
119896=0
(120584
119896)B119896(119910)E119899minus119896
(119911) (77)
The 119902-analogue looks as follows
Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider
L119873119882119860120584119902
(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
B119873119882119860120584minus119896119902
(119910) F119873119882119860120584119902
(119911)
(78)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E
119902(119905)+1)
in (68) Then
119892119899(119905) = 119891 (119905) 119892 (119905) =
(2119905)1
(E119902(1199052119902) minus 1)
(79)
This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers
Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)
be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-
mial Then119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(minus119909) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(80)
Let Φ119899119902(119909) and Ψ
119899119902(119909) be 119902-Appell polynomials Then
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910)Ψ119899minus119896119902
(119911) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902
(81)
Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E
119902(119905(119910 oplus
119902119911)) and ℎ(119905)
Or use an equivalent umbral reasoning
Corollary 27 One has the following special cases119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119909)B119902119873119882119860119899minus119896
(minus119909)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
B119873119882119860119899minus119896119902
(82)
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(83)
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(84)
8 Journal of Discrete Mathematics
Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ
119899119902(119909) and the pseudo-119902-Appell polynomial
Ψ119902
119899(119909) Then119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(119909) =
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(85)
Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and
Ψ119902
119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula
(50)
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0
=((
(
Φ0119902(119909) 0 sdot sdot sdot 0
Φ1119902(119909) Φ
0119902(119909) sdot sdot sdot 0
Φ2119902(119909) 2
119902Φ1119902(119909) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909) (119899 minus 1
1)
119902
Φ119899minus2119902
(119909) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909)
))
)
(
Ψ119902
0(119909)
minusΨ119902
1(119909)
(minus1)119899+1
Ψ119902
119899minus1(119909)
)
(86)
On the other hand by formula (49) we obtain
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot
120598W119899119902[ℎ (minus119905)]|
119905=0
=((
(
Φ0119902(0) 0 sdot sdot sdot 0
Φ1119902(0) Φ
0119902(0) sdot sdot sdot 0
Φ2119902(0) 2
119902Φ1119902(0) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(0) (119899 minus 1
1)
119902
Φ119899minus2119902
(0) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(0)
))
)
(
Ψ119902
0(0)
minusΨ119902
1(0)
(minus1)119899+1
Ψ119902
119899minus1(0)
)
(87)
The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and
(87)
We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902 (119909)B119902
NWA119899minus119896 (119909)
=
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902BNWA119899minus119896119902
(88)
Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ
119902(119909) and Ψ
119902(119909) be the 119902-Appell
vectors corresponding to 119891(119905) and 119892(119905) respectively Then
Φ119902(119909) = P
119899119902[119891 (119905)
119892 (119905)]
10038161003816100381610038161003816100381610038161003816119905=0
Ψ119902(119909) (89)
Proof By formula (49) we get
Φ119902(119909) = W
119899119902[119891 (119905)
119892 (119905)
119892 (119905)E119902(119909119905)]
10038161003816100381610038161003816100381610038161003816119905=0
= P119899119902[119891 (119905)
119892 (119905)] sdot120598W119899119902[119892 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(90)
Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials
B119873119882119860119899119902
(119909) =119899119902F119873119882119860119899minus1119902
(119909)
2
+
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119899minus119896119902
F119873119882119860119896119902
(119909)
(91)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E
119902(119905)+1)
in (89) Then
119891 (119905)
119892 (119905)=1199051
2+
1199051
E119902(119905) minus 1
(92)
Journal of Discrete Mathematics 9
We find that
(119891 (119905)
119892 (119905))
(119896)
(0) =
1
2+ BNWA119896119902 if 119896 = 1
BNWA119896119902 otherwise
(93)
The result follows by (89)
Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the
119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies
the following 119902-difference equation
119899
sum
119896=2
119902119899
120572119896minus1
119896 minus 1119902D119896119902119891 (119909119902
minus1
)
+ 119902119899
(119909119902 + 1205720)D119902119891 (119909119902
minus1
)
minus 119899 + 1119902119891 (119909) + 119902
119899
119891 (119909119902minus1
) = 0
(94)
where 120572119896equiv D119896119902(minus119891(119905)D
119902(1119891(119905)))|
119905=0
Proof We put 119891(119905) = 1119892(119905)
(
Φ1119902(119909)
Φ2119902(119909)
Φ119899119902(119909)
) = W119899119902[D119902(E119902(119909119905)
119892 (119905))]
100381610038161003816100381610038161003816100381610038161003816119905=0
= W119899119902[(119909 minus
D119902119892 (119905)
119892 (119905))E119902(119909119905)
119892 (119902119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
by (50)= P
119899119902[E119902(119909119905)
119892 (119902119905)] sdot120598W119899119902[119909 minus
D119902119892 (119905)
119892 (119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
=(((
(
Φ0119902(119909119902minus1
) 0 sdot sdot sdot 0
119902Φ1119902(119909119902minus1
) Φ0119902(119909119902minus1
) sdot sdot sdot 0
1199022
Φ2119902(119909119902minus1
) 1199022119902Φ1119902(119909119902minus1
) sdot sdot sdot 0
119902119899minus1
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909119902minus1
) 119902119899minus2
(119899 minus 1
1)
119902
Φ119899minus2119902
(119909119902minus1
) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909119902minus1
)
)))
)
(
119909+ 1205720
1205721
120572119899minus1
)
(95)
By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)
Φ120584119902(119909) = (119909 + 120572
0) 119902120584minus1
Φ120584minus1119902
(119909119902minus1
)
+
120584minus1
sum
119896=1
(120584 minus 1
119896)
119902
120572119896119902120584minus119896minus1
Φ120584minus119896minus1119902
(119909119902minus1
)
(96)
Operating with D119902on this and putting 120584 minus 1 = 119899
119899 + 1119902Φ119899119902(119909)
= 119902119899
Φ119899119902(119909119902minus1
)
+ 119902119899minus1
119899119902(119902119909 + 120572
0)Φ119899minus1119902
(119909119902minus1
)
+
119899
sum
119896=1
(119899
119896)
119902
120572119896119902119899minus119896minus1
119899 minus 119896119902
times Φ119899minus119896minus1119902
(119909119902minus1
) 119899 = 0 1 119899
(97)
where we have used the following identities
D119896119902Φ119899119902(119909119902minus1
) = 119902minus119896
119899 minus 119896 + 1119896119902Φ119899minus119896119902
D119899119902Φ119899minus1119902
(119909119902minus1
) = 0
(98)
4 Conclusion
By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]
10 Journal of Discrete Mathematics
Appendix
Bernoulli Number Formulas in Hansen
In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows
119899
sum
119896=0
(119899
119896)B119896(119910)B119899minus119896
(119911)
=
119899
sum
119896=0
(119899
119896)B119896(119910 + 119911)B
119899minus119896
(A1)
Hansen [10 page 341 50112] writes this as follows
119899
sum
119896=0
(minus1)119896(minus119899)119896
119896B119896(119910)B119899minus119896
(119911)
= 119899 (119910 + 119911 minus 1)B119899minus1
(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)
(A2)
Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials
References
[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012
[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted
[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963
[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923
[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009
[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967
[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006
[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007
[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009
[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser
2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169
Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981
[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951
[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition
[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009
[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted
[18] N E Noslashrlund Differenzenrechnung Springer 1924
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Journal of Discrete Mathematics 7
By (11) and (13) the left-hand side of (73) is equal to
(((
(
Φ0119902(119910) 0 sdot sdot sdot 0
Φ1119902(119910) Φ
0119902(119910) sdot sdot sdot 0
Φ2119902(119910) 2
119902Φ1119902(119910) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119910) (119899 minus 1
1)
119902
Φ119899minus2119902
(119910) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119910)
)))
)
times(
Ψ119902
0(119911)
Ψ119902
1(119911)
Ψ119902
119899minus1(119911)
)
(74)
By (11) the right-hand side of (73) is equal to
(
Ξ0119902(119910 ⊞119902119911)
Ξ1119902(119910 ⊞119902119911)
Ξ119899minus1119902
(119910 ⊞119902119911)
) (75)
The theorem follows by equating the last rows of formulas(75) and (74)
The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)
Corollary 24 ([11 4242]) When sum119904119897=1
119899119897= 119899
B(119899)119873119882119860119896119902
(1199091oplus119902sdot sdot sdot oplus119902119909119904)
= sum
1198981+sdotsdotsdot+119898
119904=119896
(119896
1198981 119898
119904
)
119902
119904
prod
119895=1
B(119899119895)119873119882119860119898
119895119902(119909119895)
(76)
where we assume that 119899119895operates on 119909
119895
Yang and Youn have found a new identity betweenBernoulli and Euler polynomials
2119899B119899(119910 + 119911
2)
120584
sum
119896=0
(120584
119896)B119896(119910)E119899minus119896
(119911) (77)
The 119902-analogue looks as follows
Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider
L119873119882119860120584119902
(119910 oplus119902119911) =
120584
sum
119896=0
(120584
119896)
119902
B119873119882119860120584minus119896119902
(119910) F119873119882119860120584119902
(119911)
(78)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E
119902(119905)+1)
in (68) Then
119892119899(119905) = 119891 (119905) 119892 (119905) =
(2119905)1
(E119902(1199052119902) minus 1)
(79)
This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers
Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)
be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-
mial Then119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(minus119909) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(80)
Let Φ119899119902(119909) and Ψ
119899119902(119909) be 119902-Appell polynomials Then
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910)Ψ119899minus119896119902
(119911) =
119899
sum
119896=0
(119899
119896)
119902
Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902
(81)
Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E
119902(119905(119910 oplus
119902119911)) and ℎ(119905)
Or use an equivalent umbral reasoning
Corollary 27 One has the following special cases119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119909)B119902119873119882119860119899minus119896
(minus119909)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
B119873119882119860119899minus119896119902
(82)
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(83)
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910)B119873119882119860119899minus119896119902
(119911)
=
119899
sum
119896=0
(119899
119896)
119902
F119873119882119860119896119902
(119910 oplus119902119911)B119873119882119860119899minus119896119902
(84)
8 Journal of Discrete Mathematics
Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ
119899119902(119909) and the pseudo-119902-Appell polynomial
Ψ119902
119899(119909) Then119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(119909) =
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(85)
Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and
Ψ119902
119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula
(50)
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0
=((
(
Φ0119902(119909) 0 sdot sdot sdot 0
Φ1119902(119909) Φ
0119902(119909) sdot sdot sdot 0
Φ2119902(119909) 2
119902Φ1119902(119909) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909) (119899 minus 1
1)
119902
Φ119899minus2119902
(119909) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909)
))
)
(
Ψ119902
0(119909)
minusΨ119902
1(119909)
(minus1)119899+1
Ψ119902
119899minus1(119909)
)
(86)
On the other hand by formula (49) we obtain
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot
120598W119899119902[ℎ (minus119905)]|
119905=0
=((
(
Φ0119902(0) 0 sdot sdot sdot 0
Φ1119902(0) Φ
0119902(0) sdot sdot sdot 0
Φ2119902(0) 2
119902Φ1119902(0) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(0) (119899 minus 1
1)
119902
Φ119899minus2119902
(0) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(0)
))
)
(
Ψ119902
0(0)
minusΨ119902
1(0)
(minus1)119899+1
Ψ119902
119899minus1(0)
)
(87)
The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and
(87)
We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902 (119909)B119902
NWA119899minus119896 (119909)
=
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902BNWA119899minus119896119902
(88)
Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ
119902(119909) and Ψ
119902(119909) be the 119902-Appell
vectors corresponding to 119891(119905) and 119892(119905) respectively Then
Φ119902(119909) = P
119899119902[119891 (119905)
119892 (119905)]
10038161003816100381610038161003816100381610038161003816119905=0
Ψ119902(119909) (89)
Proof By formula (49) we get
Φ119902(119909) = W
119899119902[119891 (119905)
119892 (119905)
119892 (119905)E119902(119909119905)]
10038161003816100381610038161003816100381610038161003816119905=0
= P119899119902[119891 (119905)
119892 (119905)] sdot120598W119899119902[119892 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(90)
Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials
B119873119882119860119899119902
(119909) =119899119902F119873119882119860119899minus1119902
(119909)
2
+
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119899minus119896119902
F119873119882119860119896119902
(119909)
(91)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E
119902(119905)+1)
in (89) Then
119891 (119905)
119892 (119905)=1199051
2+
1199051
E119902(119905) minus 1
(92)
Journal of Discrete Mathematics 9
We find that
(119891 (119905)
119892 (119905))
(119896)
(0) =
1
2+ BNWA119896119902 if 119896 = 1
BNWA119896119902 otherwise
(93)
The result follows by (89)
Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the
119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies
the following 119902-difference equation
119899
sum
119896=2
119902119899
120572119896minus1
119896 minus 1119902D119896119902119891 (119909119902
minus1
)
+ 119902119899
(119909119902 + 1205720)D119902119891 (119909119902
minus1
)
minus 119899 + 1119902119891 (119909) + 119902
119899
119891 (119909119902minus1
) = 0
(94)
where 120572119896equiv D119896119902(minus119891(119905)D
119902(1119891(119905)))|
119905=0
Proof We put 119891(119905) = 1119892(119905)
(
Φ1119902(119909)
Φ2119902(119909)
Φ119899119902(119909)
) = W119899119902[D119902(E119902(119909119905)
119892 (119905))]
100381610038161003816100381610038161003816100381610038161003816119905=0
= W119899119902[(119909 minus
D119902119892 (119905)
119892 (119905))E119902(119909119905)
119892 (119902119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
by (50)= P
119899119902[E119902(119909119905)
119892 (119902119905)] sdot120598W119899119902[119909 minus
D119902119892 (119905)
119892 (119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
=(((
(
Φ0119902(119909119902minus1
) 0 sdot sdot sdot 0
119902Φ1119902(119909119902minus1
) Φ0119902(119909119902minus1
) sdot sdot sdot 0
1199022
Φ2119902(119909119902minus1
) 1199022119902Φ1119902(119909119902minus1
) sdot sdot sdot 0
119902119899minus1
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909119902minus1
) 119902119899minus2
(119899 minus 1
1)
119902
Φ119899minus2119902
(119909119902minus1
) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909119902minus1
)
)))
)
(
119909+ 1205720
1205721
120572119899minus1
)
(95)
By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)
Φ120584119902(119909) = (119909 + 120572
0) 119902120584minus1
Φ120584minus1119902
(119909119902minus1
)
+
120584minus1
sum
119896=1
(120584 minus 1
119896)
119902
120572119896119902120584minus119896minus1
Φ120584minus119896minus1119902
(119909119902minus1
)
(96)
Operating with D119902on this and putting 120584 minus 1 = 119899
119899 + 1119902Φ119899119902(119909)
= 119902119899
Φ119899119902(119909119902minus1
)
+ 119902119899minus1
119899119902(119902119909 + 120572
0)Φ119899minus1119902
(119909119902minus1
)
+
119899
sum
119896=1
(119899
119896)
119902
120572119896119902119899minus119896minus1
119899 minus 119896119902
times Φ119899minus119896minus1119902
(119909119902minus1
) 119899 = 0 1 119899
(97)
where we have used the following identities
D119896119902Φ119899119902(119909119902minus1
) = 119902minus119896
119899 minus 119896 + 1119896119902Φ119899minus119896119902
D119899119902Φ119899minus1119902
(119909119902minus1
) = 0
(98)
4 Conclusion
By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]
10 Journal of Discrete Mathematics
Appendix
Bernoulli Number Formulas in Hansen
In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows
119899
sum
119896=0
(119899
119896)B119896(119910)B119899minus119896
(119911)
=
119899
sum
119896=0
(119899
119896)B119896(119910 + 119911)B
119899minus119896
(A1)
Hansen [10 page 341 50112] writes this as follows
119899
sum
119896=0
(minus1)119896(minus119899)119896
119896B119896(119910)B119899minus119896
(119911)
= 119899 (119910 + 119911 minus 1)B119899minus1
(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)
(A2)
Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials
References
[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012
[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted
[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963
[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923
[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009
[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967
[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006
[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007
[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009
[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser
2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169
Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981
[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951
[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition
[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009
[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted
[18] N E Noslashrlund Differenzenrechnung Springer 1924
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Discrete Mathematics
Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ
119899119902(119909) and the pseudo-119902-Appell polynomial
Ψ119902
119899(119909) Then119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902(119909)Ψ119902
119899minus119896(119909) =
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
Φ119896119902Ψ119899minus119896119902
(85)
Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and
Ψ119902
119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula
(50)
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0
=((
(
Φ0119902(119909) 0 sdot sdot sdot 0
Φ1119902(119909) Φ
0119902(119909) sdot sdot sdot 0
Φ2119902(119909) 2
119902Φ1119902(119909) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909) (119899 minus 1
1)
119902
Φ119899minus2119902
(119909) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909)
))
)
(
Ψ119902
0(119909)
minusΨ119902
1(119909)
(minus1)119899+1
Ψ119902
119899minus1(119909)
)
(86)
On the other hand by formula (49) we obtain
P119899119902[119891 (119905)E
119902(119909119905)] sdot
120598W119899119902[ℎ (minus119905)E
1119902(minus119909119905)]
10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot
120598W119899119902[ℎ (minus119905)]|
119905=0
=((
(
Φ0119902(0) 0 sdot sdot sdot 0
Φ1119902(0) Φ
0119902(0) sdot sdot sdot 0
Φ2119902(0) 2
119902Φ1119902(0) sdot sdot sdot 0
(119899 minus 1
0)
119902
Φ119899minus1119902
(0) (119899 minus 1
1)
119902
Φ119899minus2119902
(0) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(0)
))
)
(
Ψ119902
0(0)
minusΨ119902
1(0)
(minus1)119899+1
Ψ119902
119899minus1(0)
)
(87)
The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and
(87)
We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902 (119909)B119902
NWA119899minus119896 (119909)
=
119899
sum
119896=0
(minus1)119896
(119899
119896)
119902
BNWA119896119902BNWA119899minus119896119902
(88)
Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ
119902(119909) and Ψ
119902(119909) be the 119902-Appell
vectors corresponding to 119891(119905) and 119892(119905) respectively Then
Φ119902(119909) = P
119899119902[119891 (119905)
119892 (119905)]
10038161003816100381610038161003816100381610038161003816119905=0
Ψ119902(119909) (89)
Proof By formula (49) we get
Φ119902(119909) = W
119899119902[119891 (119905)
119892 (119905)
119892 (119905)E119902(119909119905)]
10038161003816100381610038161003816100381610038161003816119905=0
= P119899119902[119891 (119905)
119892 (119905)] sdot120598W119899119902[119892 (119905)E
119902(119909119905)]
10038161003816100381610038161003816119905=0
(90)
Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials
B119873119882119860119899119902
(119909) =119899119902F119873119882119860119899minus1119902
(119909)
2
+
119899
sum
119896=0
(119899
119896)
119902
B119873119882119860119899minus119896119902
F119873119882119860119896119902
(119909)
(91)
Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E
119902(119905)+1)
in (89) Then
119891 (119905)
119892 (119905)=1199051
2+
1199051
E119902(119905) minus 1
(92)
Journal of Discrete Mathematics 9
We find that
(119891 (119905)
119892 (119905))
(119896)
(0) =
1
2+ BNWA119896119902 if 119896 = 1
BNWA119896119902 otherwise
(93)
The result follows by (89)
Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the
119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies
the following 119902-difference equation
119899
sum
119896=2
119902119899
120572119896minus1
119896 minus 1119902D119896119902119891 (119909119902
minus1
)
+ 119902119899
(119909119902 + 1205720)D119902119891 (119909119902
minus1
)
minus 119899 + 1119902119891 (119909) + 119902
119899
119891 (119909119902minus1
) = 0
(94)
where 120572119896equiv D119896119902(minus119891(119905)D
119902(1119891(119905)))|
119905=0
Proof We put 119891(119905) = 1119892(119905)
(
Φ1119902(119909)
Φ2119902(119909)
Φ119899119902(119909)
) = W119899119902[D119902(E119902(119909119905)
119892 (119905))]
100381610038161003816100381610038161003816100381610038161003816119905=0
= W119899119902[(119909 minus
D119902119892 (119905)
119892 (119905))E119902(119909119905)
119892 (119902119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
by (50)= P
119899119902[E119902(119909119905)
119892 (119902119905)] sdot120598W119899119902[119909 minus
D119902119892 (119905)
119892 (119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
=(((
(
Φ0119902(119909119902minus1
) 0 sdot sdot sdot 0
119902Φ1119902(119909119902minus1
) Φ0119902(119909119902minus1
) sdot sdot sdot 0
1199022
Φ2119902(119909119902minus1
) 1199022119902Φ1119902(119909119902minus1
) sdot sdot sdot 0
119902119899minus1
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909119902minus1
) 119902119899minus2
(119899 minus 1
1)
119902
Φ119899minus2119902
(119909119902minus1
) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909119902minus1
)
)))
)
(
119909+ 1205720
1205721
120572119899minus1
)
(95)
By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)
Φ120584119902(119909) = (119909 + 120572
0) 119902120584minus1
Φ120584minus1119902
(119909119902minus1
)
+
120584minus1
sum
119896=1
(120584 minus 1
119896)
119902
120572119896119902120584minus119896minus1
Φ120584minus119896minus1119902
(119909119902minus1
)
(96)
Operating with D119902on this and putting 120584 minus 1 = 119899
119899 + 1119902Φ119899119902(119909)
= 119902119899
Φ119899119902(119909119902minus1
)
+ 119902119899minus1
119899119902(119902119909 + 120572
0)Φ119899minus1119902
(119909119902minus1
)
+
119899
sum
119896=1
(119899
119896)
119902
120572119896119902119899minus119896minus1
119899 minus 119896119902
times Φ119899minus119896minus1119902
(119909119902minus1
) 119899 = 0 1 119899
(97)
where we have used the following identities
D119896119902Φ119899119902(119909119902minus1
) = 119902minus119896
119899 minus 119896 + 1119896119902Φ119899minus119896119902
D119899119902Φ119899minus1119902
(119909119902minus1
) = 0
(98)
4 Conclusion
By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]
10 Journal of Discrete Mathematics
Appendix
Bernoulli Number Formulas in Hansen
In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows
119899
sum
119896=0
(119899
119896)B119896(119910)B119899minus119896
(119911)
=
119899
sum
119896=0
(119899
119896)B119896(119910 + 119911)B
119899minus119896
(A1)
Hansen [10 page 341 50112] writes this as follows
119899
sum
119896=0
(minus1)119896(minus119899)119896
119896B119896(119910)B119899minus119896
(119911)
= 119899 (119910 + 119911 minus 1)B119899minus1
(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)
(A2)
Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials
References
[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012
[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted
[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963
[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923
[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009
[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967
[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006
[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007
[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009
[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser
2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169
Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981
[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951
[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition
[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009
[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted
[18] N E Noslashrlund Differenzenrechnung Springer 1924
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Discrete Mathematics 9
We find that
(119891 (119905)
119892 (119905))
(119896)
(0) =
1
2+ BNWA119896119902 if 119896 = 1
BNWA119896119902 otherwise
(93)
The result follows by (89)
Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the
119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies
the following 119902-difference equation
119899
sum
119896=2
119902119899
120572119896minus1
119896 minus 1119902D119896119902119891 (119909119902
minus1
)
+ 119902119899
(119909119902 + 1205720)D119902119891 (119909119902
minus1
)
minus 119899 + 1119902119891 (119909) + 119902
119899
119891 (119909119902minus1
) = 0
(94)
where 120572119896equiv D119896119902(minus119891(119905)D
119902(1119891(119905)))|
119905=0
Proof We put 119891(119905) = 1119892(119905)
(
Φ1119902(119909)
Φ2119902(119909)
Φ119899119902(119909)
) = W119899119902[D119902(E119902(119909119905)
119892 (119905))]
100381610038161003816100381610038161003816100381610038161003816119905=0
= W119899119902[(119909 minus
D119902119892 (119905)
119892 (119905))E119902(119909119905)
119892 (119902119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
by (50)= P
119899119902[E119902(119909119905)
119892 (119902119905)] sdot120598W119899119902[119909 minus
D119902119892 (119905)
119892 (119905)]
100381610038161003816100381610038161003816100381610038161003816119905=0
=(((
(
Φ0119902(119909119902minus1
) 0 sdot sdot sdot 0
119902Φ1119902(119909119902minus1
) Φ0119902(119909119902minus1
) sdot sdot sdot 0
1199022
Φ2119902(119909119902minus1
) 1199022119902Φ1119902(119909119902minus1
) sdot sdot sdot 0
119902119899minus1
(119899 minus 1
0)
119902
Φ119899minus1119902
(119909119902minus1
) 119902119899minus2
(119899 minus 1
1)
119902
Φ119899minus2119902
(119909119902minus1
) sdot sdot sdot (119899 minus 1
119899 minus 1)
119902
Φ0119902(119909119902minus1
)
)))
)
(
119909+ 1205720
1205721
120572119899minus1
)
(95)
By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)
Φ120584119902(119909) = (119909 + 120572
0) 119902120584minus1
Φ120584minus1119902
(119909119902minus1
)
+
120584minus1
sum
119896=1
(120584 minus 1
119896)
119902
120572119896119902120584minus119896minus1
Φ120584minus119896minus1119902
(119909119902minus1
)
(96)
Operating with D119902on this and putting 120584 minus 1 = 119899
119899 + 1119902Φ119899119902(119909)
= 119902119899
Φ119899119902(119909119902minus1
)
+ 119902119899minus1
119899119902(119902119909 + 120572
0)Φ119899minus1119902
(119909119902minus1
)
+
119899
sum
119896=1
(119899
119896)
119902
120572119896119902119899minus119896minus1
119899 minus 119896119902
times Φ119899minus119896minus1119902
(119909119902minus1
) 119899 = 0 1 119899
(97)
where we have used the following identities
D119896119902Φ119899119902(119909119902minus1
) = 119902minus119896
119899 minus 119896 + 1119896119902Φ119899minus119896119902
D119899119902Φ119899minus1119902
(119909119902minus1
) = 0
(98)
4 Conclusion
By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]
10 Journal of Discrete Mathematics
Appendix
Bernoulli Number Formulas in Hansen
In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows
119899
sum
119896=0
(119899
119896)B119896(119910)B119899minus119896
(119911)
=
119899
sum
119896=0
(119899
119896)B119896(119910 + 119911)B
119899minus119896
(A1)
Hansen [10 page 341 50112] writes this as follows
119899
sum
119896=0
(minus1)119896(minus119899)119896
119896B119896(119910)B119899minus119896
(119911)
= 119899 (119910 + 119911 minus 1)B119899minus1
(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)
(A2)
Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials
References
[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012
[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted
[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963
[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923
[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009
[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967
[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006
[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007
[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009
[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser
2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169
Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981
[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951
[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition
[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009
[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted
[18] N E Noslashrlund Differenzenrechnung Springer 1924
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Discrete Mathematics
Appendix
Bernoulli Number Formulas in Hansen
In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows
119899
sum
119896=0
(119899
119896)B119896(119910)B119899minus119896
(119911)
=
119899
sum
119896=0
(119899
119896)B119896(119910 + 119911)B
119899minus119896
(A1)
Hansen [10 page 341 50112] writes this as follows
119899
sum
119896=0
(minus1)119896(minus119899)119896
119896B119896(119910)B119899minus119896
(119911)
= 119899 (119910 + 119911 minus 1)B119899minus1
(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)
(A2)
Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials
References
[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012
[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted
[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963
[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923
[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009
[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967
[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006
[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007
[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009
[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser
2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169
Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981
[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951
[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition
[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009
[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted
[18] N E Noslashrlund Differenzenrechnung Springer 1924
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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