a study on a new class of q-bernoulli, q-euler and q
TRANSCRIPT
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A Study on A New Class of q-Bernoulli, q-Euler
and q-Genocchi Polynomials
Mohammad Momenzadeh
Submitted to the
Institute of Graduate Studies and Research
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Applied Mathematics and Computer Science
Eastern Mediterranean University
February 2016
GazimaÄusa, North Cyprus
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ABSTRACT
This thesis is aimed to study a new class of Bernoulli, Euler and Genocchi
polynomials in the means of quantum forms. To achieve this aim, we introduce a
new class of q-exponential function and using properties of this function; we reach to
the interesting formulae. The q-analogue of some familiar relations as an addition
theorem for these polynomials is found. Explicit relations between these classes of
polynomials are given. In addition, the new differential equations related to these
polynomials are studied. Moreover, improved q-exponential function creates a new
class of q-Bernoulli numbers and like the ordinary case, all the odd coefficient
becomes zero and leads us to the relation of these numbers and q-trigonometric
functions. At the end we introduce a unification form of q-exponential function. In
this way all the properties of these kinds of polynomials investigated in a general
case. We also focus on two important properties of q-exponential function that lead
us to the symmetric form of q- Euler, q-Bernoulli and q-Genocchi numbers. These
properties and the conditions of them are studied.
Keywords: q-Exponential Function, q-Calculus, q-Polynomials, q-Bernoulli, q-
Euler, q-Genocchi, q-Trigonometric Functions.
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ÃZ
Bu tez kuantum formları uasıtası ile Bernoulli, Euler ve Gennochi polinomların yeni
bir sınıfını incelemeyi amaçlamaktadır. Bu amaca ulaÅmak için, q-ÃŒstel fonksiyonları
ve bu fonksiyonların özellikleri kullanılarak yeni bir sınıf tanıtılmıÅtır.Bu polinomlar
için bazı bilindik ilıskilerin q-uyarlamları ek teorem olarak bulunmuÅtur.
Bu tÃŒr polinom sınıfları arasındaki kapalı iliÅkiler verilmiÅtir. Ayrıca bu polinomlarla
iliÅkili yeni diferansiyel denklemler çalıÅılmıÅtır. GeliÅtirilmiÅ q-ÃŒstel
fonksiyonlarnin yeni bir q-Bernoulli sayısı oluÅturduÄu da gösterilmiÅtir. Buna göre
bilinen q-1 durumunda oldÌgu gibi tÌm tek katsayların sıfır olarak q-Ìstel
fonksiyonları, için birleÅme formu, elde edilmiÅtir. Böylelıkle, bu tÃŒr polinomların
tÃŒm özellikleri genelleÅtirilmiÅtir. Ayrıca, q-Euler, q-Bernoulli ve q-Gennochi
sayılaının simetri formlarını elde etmemızi saÄloyon, iki önemli q-ÃŒstel fonksiyon
özelliÄine de odaklanılmıÅtır.
Anahtar Kelimeler: q-Ãstel Fonksiyonlar, q-KalkÃŒlÃŒs, q-Polinomlar, q-Bernoulli,
q-Euler, q-Genocchi, q-Trigonometrik Fonksiyonla.
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To the spirit of my father
To my compassionate unique mother
And
To my lovely Brother and sisters
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ACKNOWLEDGMENT
I would like to thank my kind and nice supervisor Prof. Dr. Nazim I. Mahmudov,
who inspired my creativity and changed my view to the life and specifically to
Mathematics, because of this continuous support, warm help, innovative suggestions,
and unforgettable encouragement during preparing the scientific concepts of this
manuscript. Also, I would like to thank Prof. Dr. Agamirza Bashirov, assoc. Prof. Dr.
Sonuc Zorlu Ogurlu, Prof. Dr. Rashad Aliev and assoc. Prof. Dr. Muge saadatoglu
for their kind help during the period of my PhD studies. Thanks to them all the
people who make such a good atmosphere in the department of mathematics at
EMU. I would like to thank my family because of their strong patience, selfless help
and unlimited compassion. I wish to thank my kind brother Kamal Momemzadeh and
my lovely sister Mandana Momenzadeh for their remarkable understanding and pure
love. The last, but not the least, I wish to thank my friends, Parandis Razani, Alireza
Kordjamshidi, Alireza Zarrin and Elnaz Rajabpour for their indeterminable kind and
support.
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TABEL OF CONTENTS
ABSTRACT .......................................................................................................................... iii
ÃZ ........................................................................................................................................... iv
DEDICATION ....................................................................................................................... v
ACKNOWLEDGMENT ..................................................................................................... vi
1 INTRODUCTION ............................................................................................................. 1
1.1 Classical Bernoulli Numbers ..................................................................................... 1
1.2 Quantum Calculus and q-Exponential Function ...................................................... 5
2 PRELIMIMARY AND DEFINITIONS ......................................................................... 7
2.1 Definitions and Notations ........................................................................................... 7
2.2 Improved q-Exponential Function .......................................................................... 10
2.3The New Class of q-Polynomials ............................................................................. 12
3 APPROACH TO THE NEW CLASS OF q-BERNOULLI, q-EULER AND q-
GENNOCHI POLYNOMIALS ........................................................................................ 15
3.1 Relations to The q-Trigonometric Functions ......................................................... 15
3.2 Addition and Difference Equations and Corollaries ............................................. 18
3.3 Differential Equations Related to q-Bernoulli Polynomials ................................ 24
3.4 Explicit Relationship Between q-Bernoulli and q-Euler Polynomials ............... 26
4 UNIFICATION OF q-EXPONENTIAL FUNCTION AND RELATED
POLYNOMIALS ................................................................................................................ 33
4.1 Preliminary Results ................................................................................................... 33
4.2 New Exponential Function and Its Properties ....................................................... 36
4.3 Related q-Bernoulli Polynomial .............................................................................. 40
4.4 Unification of q-numbers ......................................................................................... 41
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REFRENCES ....................................................................................................................... 45
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Chapter 1
INTRODUCTION
1.1 Classical Bernoulli Numbers
Two thousand years ago, Greek mathematician Pythagoras noted about triangle
numbers that is 1 + 2 +â¯+ ð. after that time, Archimedes proposed
12 + 22 +â¯+ ð2 =1
6ð(ð + 1)(2ð + 1). (1.1.1)
Later, Aryabhata, The Indian mathematician, found out
13 + 23 +â¯+ ð3 = (1
2ð(ð + 1))
2
. (1.1.2)
But Jacobi was the first who gave the vigorous proof in 1834. AL-Khwarizm, the
Arabian mathematician found this summationâs result for the higher power. He
showed that
14 + 24 +â¯+ ð4 =1
30ð(ð + 1)(2ð + 1)(3ð2 + 3ð â 1). (1.1.3)
The more generalized formula for â ðððð=1 , where r is any natural number is studied
in the last few centuries. Among this investigation, the Bernoulli numbers are much
significant. In this chapter, we present an elementary examination of the
development of Bernoulli numbers throughout the time. We also aim to explore its
properties and its application in other fields of mathematics. In addition, by
introducing the generating function, we will take a look to the exponential function
and its generalization to the quantum form [12].
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Swiss mathematician Jacob Bernoulli (1654-1705), was the person who found the
general formula for this kind of the summation. He found that [3]
ðð(ð) =â ðððâ1
ð=1
=âðµðð!
ð
ð=0
ð!
(ð â ð + 1)!ððâð+1. (1.1.4)
The ðµðâs numbers here are independent of r and named Bernoulli numbers. The first
few Bernoulliâs numbers are as following:
ðµ0 = 1, ðµ1 = â1
2, ðµ2 =
1
6, ðµ3 = 0, ðµ4 = â
1
30, ðµ5 = 0, ðµ6 =
1
42,âŠ
It is tempting to guess that |ðµð|â 0 as ð â â. However, if we consider some other
numbers in the sequence,
ðµ8 = â1
30, ðµ10 =
5
66, ðµ12 = â
691
2730, ðµ14 =
7
6, ðµ16 = â
3617
510, âŠ
We notice their values are general growing with alternative sign.
An equivalent definition of the Bernoulliâs numbers is obtained by using the series
expansion as a generating function
ð¥
ðð¥ â 1=â
ðµðð¥ð
ð!
â
ð=0
. (1.1.5)
In fact, by writing the Taylor expansion for ððð¡ and adding them together, we have:
â ðð(ð)ð¡ð
ð!
â
ð=0
= 1+ ðð¡ + ð2ð¡ +â¯+ ð(ðâ1)ð¡ =1â ððð¡
1 â ðð¡=1â ððð¡
ð¡Ã
ð¡
1 â ðð¡
= (â ððð¡ðâ1
ð!
â
ð=1
)(âðµðð¡ð
ð!
â
ð=0
)
= â (â(ð+ 1ð
)
ð
ð=0
ðµðððâð+1)
ð¡ð
ð!
1
ð+ 1
â
ð=0
.
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This calculation, lead us to (1.1.4). This technique of proof is used temporary. In
addition, this definition of Bernoulliâs number connects them to the trigonometric
function as following:
ð¥
ðð¥ â 1+ð¥
2=ð¥
2(
2
ðð¥ â 1+ 1) =
ð¥
2
ðð¥ + 1
ðð¥ â 1=ð¥
2ð¶ðð¡â (
ð¥
2). (1.1.6)
It can be rewritten
ð¥
2ð¶ðð¡â (
ð¥
2) =â
ðµ2ðð¥2ð
(2ð)!.
â
ð=0
(1.1.7)
If we substitute ð¥ with 2ðð¥, then it gives
ð¥ð¶ðð¡(ð¥) = â(â1)ððµ2ð(2ð¥)
2ð
(2ð)!
â
ð=0
ð¥ â [âð, ð]. (1.1.8)
The following identities can be written
ð¡ððâ(ð¥) = â2(4ð â 1)ðµ2ð(2ð¥)
2ðâ1
(2ð)!
â
ð=1
ð¥ â (âð
2,ð
2), (1.1.9)
ð¡ðð(ð¥) = â(â1)ð2(1 â 4ð)ðµ2ð(2ð¥)
2ðâ1
(2ð)!
â
ð=1
ð¥ â (âð
2,ð
2). (1.1.10)
Another equivalent definition of Bernoulli number, which is useful, is the recurrence
formula for these numbers
{â (
ðð)ðµð = 0, ð ⥠2.
ðâ1
ð=0
ðµ1 = 1
(1.1.11)
This definition can be found easily from the series definition, by writing the Taylor
expansion for exponential function and using the Cauchy product. This formula can
be used to evaluate the Bernoulli numbers inductively (see [5] ).As we saw, the
exponential function made a main role in these definitions, and by changing the
alternative functions; we can reach to a new definitions. This idea was leading a lot
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of mathematician to find a new version of the Bernoulli numbers; we also did one of
them.
The Bernoulli polynomials are defined in the means of Taylor expansion as
following
ð§ðð§ð¥
ðð§ â 1=â
ðµð(ð¥)
ð!
â
ð=0
ð§ð. (1.1.12)
For each nonnegative integer ð, these ðµð(ð¥), are the polynomials with respect to ð¥. Taking
derivative on both sides of (1.1.12) a derivative with respect to ð¥, we get
âðµâ²ð(ð¥)
ð!
â
ð=0
ð§ð = ð§ð§ðð§ð¥
ðð§ â 1 = â
ðµð(ð¥)
ð!
â
ð=0
ð§ð+1. (1.1.13)
Equating coefficient of ð§ð, where ð ⥠1, leads us to another important identity
ðµâ²ð(ð¥) = ððµðâ1(ð¥). (1.1.14)
The fact ðµ0(ð¥) = 1, can be yield by tending ð§ to zero at (1.1.13). This and the above
identity together, show that ðµð(ð¥) is a polynomial in degree of ð and it begins with
the coefficient that is unity. If we use this identity and know what the constant terms
are, then we could evaluate ðµð(ð¥) (Bernoulli polynomials) one by one. It is clear
that, by putting ð¥ = 0, we reach to the Bernoulliâs numbers. There are too many
interesting properties for this polynomials and numbers, which are studied in the last
few centuries. The application of them is going forward to the many branches of
mathematics and physics, as combinatorics, theory of numbers, quantum
information, etc. (see [20], [24]) We will list some of these properties without proof.
The proofs can be found at [14].
ðµð(ð¥ + 1)âðµð(ð¥) = ðð¥ðâ1, (1.1.15)
ðµð(1 â ð¥) = (â1)ððµð(ð¥), (1.1.16)
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ðµð =â1
ð + 1(â(â1)ð (ð
ð) ðð
ð
ð=0
)
ð
ð=0
, (1.1.17)
ð! =â(â1)ð+ð (ðð)ðµð(ð)
ð
ð=0
, (1.1.18)
ð(ð ) = â1
ðð
â
ð=1
â 2ð(2ð) = (â1)ð+1(2ð)2ð
(2ð)!ðµ2ð. (1.1.19)
Where ð(ð ) is known as Riemann-zeta function at (1.1.19).
1.2 Quantum Calculus and q-Exponential Function
The fascinate world of quantum calculus has been started by the definition of
derivative, where the limit has not been taken.
Consider the derivative expression without any limit, as ð(ð¥)âð(ð¥0)
ð¥âð¥0. The familiar
definition of derivative ðð
ðð¥ of a function ð(ð¥) at ð¥ = ð¥0 can be yield by tending ð¥
to ð¥0. However, if we take ð¥ = ðð¥0 (where ð â 1 and is a fixed number) and do not
take a limit, we will arrive at the world of quantum calculus. The corresponding
expression is a definition of the q-derivative of ð(ð¥). The theory of quantum calculus
can be traced back at a century ago to Euler and Gauss [7] [2]. Moreover, the
significant contributions of Jackson made a big role [9]. Recently in these days, a lot
of scientifics are working in this field to develop and apply the q-calculus in
mathematical physics, especially concerning quantum mechanics [18] and special
functions [10], many papers were mentioned the various models of elementary
functions, including trigonometric functions and exponential by deforming formula
of the functions in the means of quantum calculus [1]. For instance, many notions
and results is discovered along the traditional lines of ordinary calculus, the q-
derivative of ð¥ð becomes [ð]ðð¥ðâ1, where [ð]ð =
ððâ1
ðâ1. This q-analogue of the
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number helps us to define the new version of familiar functions such an exponential
one. We redefine these functions by their Taylor expansion and the different
notation. In this case, two kind of q-exponential functions are defined as follows
ððð¥ =â
ð¥ð
[ð]ð!=â
1
(1 â (1 â ð)ððð¥)
â
ð=0
0 < |ð| < 1, |ð¥| <1
|1 â ð|,
â
ð=0
(1.2.1)
ðžðð¥ =â
ðð(ðâ1)2 ð¥
ð
[ð]ð!=â(1 + (1 â ð)ððð¥) 0 < |ð| < 1, ð¥ â â.
â
ð=0
â
ð=0
(1.2.2)
These definitions are coming from q-Binomial theorem that will discuss in the next
chapter.
Many mathematicians encourage defining the q-functions for the preliminary
functions, such that, they coincide with the classic properties of them (1.2.1) and
(1.2.2) identities are discovered by Euler. In general ððð¥ðððŠ â ðð
ð¥+ðŠ. But additive
property of the q-exponentials holds true if ð¥ and ðŠ are not commutative i.e. ðŠð¥ =
ðð¥ðŠ. It is rarely happened, and make a lot of restrictions to use them. So, we used the
improved exponential function as follows [6]
ððð¥ = ðð
ð¥2ðžð
ð¥2 =â
ð¥ð(1 + ð)âŠ(1+ ððâ1)
2ðâ1[ð]ð!
=â(1 + (1 â ð)ðð
ð¥2)
(1 â (1 â ð)ððð¥2)
â
ð=0
â
ð=0
(1.2.3)
For the remained, we assume that 0 < ð < 1. The improved q-exponential function
is analytic in the disk |ð§| <2
1âð. The important property of this function is
ððâð§ =
1
ððð§ , |ðð
ðð¥| = 1 ð§ â â, ð¥ â â (1.2.4)
At the end, we will unify the q-exponential functions and investigate the properties
of this general case. In addition q-analogue of some classical relations will be given.
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Chapter 2
PRELIMINARY AND DEFINITIONS
2.1 Definitions and Notations
In this section we introduce some definitions and also some related theorem about q-
calculus. Base of these information are [7], [14] and [24]. All of these definitions
and notations can be found there.
Definition 2.1. Let us assume that ð(ð¥) is an arbitrary function, then q-differential is
defined by the following expression
ðð(ð(ð¥)) = ð(ðð¥)â ð(ð¥). (2.1.1)
We can call the following expression by q-derivative of ð(ð¥)
ð·ð(ð(ð¥)) =ðð(ð(ð¥))
ððð¥=ð(ðð¥)â ð(ð¥)
(ð â 1)ð¥ 0 â ð¥ â â, |ð| â 1 (2.1.2)
Note that limðâ1â ð·ð(ð(ð¥)) =ðð(ð¥)
ðð¥, where ð(ð¥) is a differentiable function. The q-
analogue of product rule and quotient rule can be demonstrated by
ð·ð(ð(ð¥)ð(ð¥)) = ð(ðð¥)ð·ð(ð(ð¥))+ð(ð¥)ð·ð(ð(ð¥)), (2.1.3)
ð·ð (ð(ð¥)
ð(ð¥)) =
ð(ð¥)ð·ð(ð(ð¥))â ð(ð¥)ð·ð(ð(ð¥))
ð(ð¥)ð(ðð¥). (2.1.4)
By symmetry, we can interchange ð and ð, and obtain another form of these
expressions as well. Let us introduce the q-numberâs notation
[ð]ð =ðð â 1
ð â 1= ððâ1 +â¯+ 1, ð â â\{1}, ð â â, ðð â 1. (2.1.5)
In a natural way [ð]ð! can be defined by
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[0]ð! = 1, [ð]ð! = [ð â 1]ð! [ð]ð. (2.1.6)
Remark 2.2. It is clear that
limðâ1
[ð]ð = ð, limðâ1
[ð]ð! = ð! . (2.1.7)
Definition 2.3. For any complex number ð, we can define q-shifted factorial
inductively as following
(ð; ð)0 = 1, (ð; ð)ð = (ð; ð)ðâ1(1 â ððâ1ð), ð â â. (2.1.8)
and in a case that ð â â, we have
(ð; ð)â =â(1 â ððð)
â
ð=0
, |ð| < 1, ð â â. (2.1.9)
Definition 2.4. The q-binomial coefficient can be defined as follows
[ðð]ð
=[ð]ð!
[ð]ð! [ð â ð]ð!, ð, ð â â. (2.1.10)
We can present q-binomial coefficient by q-shifted factorial
[ðð]ð
=(ð; ð)ð
(ð; ð)ðâð(ð; ð)ð (2.1.11)
Theorem 2.5. Suppose that ð is a real function on a closed interval [ð, ð], ð is a
positive integer, ð(ð)(ð¥) exists for every ð¥ â (ð, ð) and ð(ðâ1) is continuous on this
interval. Let ðŒ, ðœ be two distinct numbers of this interval, and define
ð(ð¥) = âð(ð)(ðŒ)
ð!(ð¥ â ðŒ)ð
ðâ1
ð=0
, (2.1.12)
Thus there exist a number ðŠ, such that ðŒ < ðŠ < ðœ and
ð(ðœ) = ð(ðœ) +ð(ð)(ðŠ)
ð!(ðœ â ðŒ)ð. (2.1.13)
Remark 2.6. The previous theorem is Taylor theorem [21], the general form of
theorem is presented next [24]. By using the next theorem we lead to the definitions
of classical q-exponential functions.
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Theorem 2.7. Let ð be an arbitrary number and ð· is defined as a linear operator on
the space of polynomials and {ð0(ð¥), ð1(ð¥), ð2(ð¥), ⊠} be a sequence of polynomials
satisfying three conditions:
1) ð0(ð) = 1 and ðð(ð) = 0 for any ð ⥠1;
2) Degree of ðð(ð¥) is equal to ð
3) For any ð ⥠1, ð·(1) = 0 and ð·(ðð(ð¥)) = ððâ1(ð¥) .
Then, for any polynomial ð(ð¥) of degree ð, one has the following generalized
Taylor formula:
ð(ð¥) = â(ð·ðð)(ð)ðð(ð¥).
ð
ð=0
(2.1.14)
Definition 2.8. The q-analogue of ð-th power of (ð¥ + ð) is (ð¥ + ð)ðð and defined by
the following expression
(ð¥ + ð)ðð = {
1 ðð ð = 0
(ð¥ + ð)(ð¥ + ðð)⊠(ð¥ + ððâ1ð) ðð ð ⥠1. (2.1.15)
Corollary 2.9. Gaussâs binomial formula can be presented by
(ð¥ + ð)ðð =â[
ðð ]ð
ð
ð=0
ðð(ðâ1)
2â ððð¥ðâð. (2.1.16)
Definition 2.10. Suppose that 0 < ð < 1. If for some 0 †ð < 1, value of |ð(ð¥)ð¥ð|
is bounded on the interval (0, ðµ], then the following integral is defined by the series
that converged to a function ð¹(ð¥) on (0, ðµ] and is called Jackson integral
â«ð(ð¥)ððð¥ = (1 â ð)ð¥âððð(
â
ð=0
ððð¥). (2.1.17)
Definition 2.11. Classical q-exponential functions are defined by Euler [24]
ðð(ð¥) = âð¥ð
[ð]ð!
â
ð=0
=â1
(1 â (1 â ð)ððð¥), 0 < |ð| < 1, |ð¥| <
1
|1 â ð|
â
ð=0
,
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10
ðžð(ð¥) = âð¥ðð
ð(ðâ1)2â
[ð]ð!
â
ð=0
=â(1 + (1 â ð)ððð¥), 0 < |ð| < 1, ð¥ â â
â
ð=0
.
Proposition 2.12. Some properties of q-exponential functions are listed as follow
(a) (ðð(ð¥))â1
= ðžð(ð¥), ð1ð
(ð¥) = ðžð(ð¥),
(b) ð·ððžð(ð¥) = ðžð(ðð¥), ð·ððð(ð¥) = ðð(ð¥),
(c) ðð(ð¥ + ðŠ) = ðð(ð¥). ðð(ðŠ) ðð ðŠð¥ = ðð¥ðŠ
Definition 2.13. These q-exponential functions that defined at 2.11 generate two pair
of the q-trigonometric functions. We have
ð ððð(ð¥) =ðð(ðð¥) â ðð(âðð¥)
2ð, ðððð(ð¥) =
ðžð(ðð¥) â ðžð(âðð¥)
2ð
ððð ð(ð¥) =ðð(ðð¥) + ðð(âðð¥)
2, ð¶ðð ð(ð¥) =
ðžð(ðð¥) + ðžð(âðð¥)
2.
Proposition 2.14. We can easily derive some properties of standard q-trigonometric
functions by taking into account the properties of q-exponential function
(a) ððð ð(ð¥)ð¶ðð ð(ð¥) + ð ððð(ð¥)ðððð(ð¥) = 1,
(b) ð¶ðð ð(ð¥)ð ððð(ð¥) = ððð ð(ð¥)ðððð(ð¥),
(c) ð·ð (ð ððð(ð¥)) = ððð ð(ð¥), ð·ð (ððð ð(ð¥)) = âð ððð(ð¥),
(d) ð·ð (ðððð(ð¥)) = ð¶ðð ð(ðð¥), ð·ð (ð¶ðð ð(ð¥)) = âðððð(ðð¥).
Remark 2.15. The corresponding tangents and cotangents coincide ð¡ððð(ð¥) =
ðððð(ð¥), ððð¡ð(ð¥) = ð¶ðð¡ð(ð¥).
2.2 Improved q-Exponential Function
Definition 2.16. Improved q-exponential function is defined as follows
ðð(ð¥) â ðð (ð¥
2)ðžð (
ð¥
2) = â
1+ (1 â ð)ððð¥2
1 â (1 â ð)ððð¥2
,
â
ð=0
(2.2.1)
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11
Theorem 2.17. The q-exponential function ðð(ð¥) which is defined at (2.2.1) is
analytic in the disk |ð¥| < ð ð, where ð ð is as follows
ð ð =
{
2
1 â ð ðð 0 < ð < 1
2ð
1 â ð ðð ð > 1
â ðð ð = 1.
Moreover, we can write the following expansion for ðð(ð¥) [6]
ðð(ð¥) = âð¥ð
[ð]ð!
(â1; ð)ð2ð
â
ð=0
. (2.2.2)
Theorem 2.18. For ð§ â â , ð¥ â â, improved q-exponential function ðð(ð§), has the
following property
(a) ( ðð(ð§))â1
= ðð(âð§), | ðð(ðð¥)| = 1,
(b) ðð(ð§) = ð1ð
(ð§), ð·ð (ðð(ð§)) = ðð(ð§)+ ðð(ðð§)
2.
Remark 2.19. As we mentioned it before, in a general case ðð(ð¥ + ðŠ) â
ðð(ð¥). ðð(ðŠ). One of the advantages of using improved q-exponential is the property
(a) at the previous theorem. The form of improved q-exponential, motivates us to
define the following
Definition 2.20. The q-addition and q-subtraction can be defined as follow
(ð¥âšððŠ)ðââ[
ðð]ð
ð
ð=0
(â1; ð)ð(â1; ð)ðâð2ð
ð¥ððŠðâð, ð = 0,1,2, âŠ, (2.2.3)
(ð¥ âð ðŠ)ðââ[
ðð]ð
ð
ð=0
(â1; ð)ð(â1; ð)ðâð2ð
ð¥ð(âðŠ)ðâð, ð = 0,1,2, âŠ. (2.2.4)
The direct consequence of this definition is the following identity
ðð(ð¡ð¥) ðð(ð¡ðŠ) = â(ð¥âšððŠ)ð ð¡ð
[ð]ð!
â
ð=0
. (2.2.5)
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Remark 2.21. The properties that mentioned at (2.18) encourage some
mathematicians to use this improved q-exponential in their works [11], [17].
Recently, we are working on other applications of improved q-exponential.
2.3 The New Class of q-Polynomials
In this section, we study a new class of q-polynomials including q-Bernoulli, q-Euler
and q-Genocchi polynomials. First, we discuss about the classic definitions of them.
Definition 2.22. Classic Bernoulli, Euler and Genocchi polynomials can be defined
by their generating functions as following. We named them ðµð(ð¥), ðžð(ð¥) and ðºð(ð¥)
repectively.
ð¡ðð¡ð¥
ðð¡ â 1= âðµð(ð¥)
ð¡ð
ð!
â
ð=0
,2ðð¡ð¥
ðð¡ + 1=âðžð(ð¥)
ð¡ð
ð!
â
ð=0
,2ð¡ðð¡ð¥
ðð¡ + 1=âðºð(ð¥)
ð¡ð
ð!
â
ð=0
.
If we put ð¥ = 0, we have ðµð(0) = ðð, ðžð(0) = ðð, ðºð(0) = ðð, which we call
them Bernoulli numbers, Euler numbers and Genocchi numbers respectively. Also
we can define them by the recurrence formula, which is coming from the Cauchy
product of both sides of generating function. Main idea of introducing the new
version of these polynomials is coming from these definitions. The q-Bernoulli
numbers and polynomials have been studied by Carlitz, when he modified the form
of recurrence formula. [4] Srivastava and PintÂŽer demonstrated a few theorems and
they found the explicit relations that exist between the Euler polynomials and
Bernoulli polynomials in [19]. Also they generalize some of these polynomials.
Some properties of the q-analogues of Bernoulli polynomials and Euler polynomials
and Genocchi polynomials are found by Kim et al. In [23], [13]. Some recurrence
relations were given in these papers. In addition, the extension of Genocchi numbers
in the means of quantum is studied by Cenkci et al. in a different manner at [15].
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Definition 2.23. Assume that ð is a complex number that 0 < |ð| < 1. Then we can
define q-Bernoulli numbers ðð,ð and polynomials ðµð,ð(ð¥, ðŠ) as follows by using
generating functions
ðµï¿œÌï¿œ âð¡ ðð (â
ð¡2)
ðð (ð¡2) â ðð (â
ð¡2)=
ð¡
ðð(ð¡) â 1= âðð,ð
ð¡ð
[ð]ð!
â
ð=0
ð€âððð |ð¡|
< 2ð,
(2.3.1)
ð¡ ðð(ð¡ð¥) ðð(ð¡ðŠ)
ðð(ð¡) â 1= âðµð,ð(ð¥, ðŠ)
ð¡ð
[ð]ð!
â
ð=0
ð€âððð |ð¡| < 2ð. (2.3.2)
Definition 2.24. Assume that ð is a complex number that 0 < |ð| < 1. Then we can
define q-Euler numbers ðð,ð and polynomials ðžð,ð(ð¥, ðŠ) as follows by using
generating functions
ðžï¿œÌï¿œ â2 ðð (â
ð¡2)
ðð (ð¡2) + ðð (â
ð¡2)=
2
ðð(ð¡) + 1= âðð,ð
ð¡ð
[ð]ð!
â
ð=0
ð€âððð |ð¡| < ð, (2.3.2)
2 ðð(ð¡ð¥) ðð(ð¡ðŠ)
ðð(ð¡) + 1= âðžð,ð(ð¥, ðŠ)
ð¡ð
[ð]ð!
â
ð=0
ð€âððð |ð¡| < 2ð. (2.3.3)
Definition 2.25 Assume that ð is a complex number that 0 < |ð| < 1. Then in a
same way, we can define q-Genocchi numbers ðð,ð and polynomials ðºð,ð(ð¥, ðŠ) as
follows by using generating functions
ðºï¿œÌï¿œ â2ð¡ ðð (â
ð¡2)
ðð (ð¡2) + ðð (â
ð¡2)=
2ð¡
ðð(ð¡) + 1= âðð,ð
ð¡ð
[ð]ð!
â
ð=0
ð€âððð |ð¡| < ð, (2.3.4)
2ð¡ ðð(ð¡ð¥) ðð(ð¡ðŠ)
ðð(ð¡) + 1= âðºð,ð(ð¥, ðŠ)
ð¡ð
[ð]ð!
â
ð=0
ð€âððð |ð¡| < ð. (2.3.5)
Definition 2.26. Like the previous definitions, we will assume that ð is a complex
number that 0 < |ð| < 1. The q-tangent numbers ðð,ð can be defined as following
by using generating functions as following
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14
ð¡ððâðð¡ = ðð¡ððð(ðð¡) = ðð(ð¡) â ðð(âð¡)
ðð(ð¡) + ðð(âð¡)= ðð(2ð¡) â 1
ðð(2ð¡) + 1= âð2ð+1,ð
(â1)ðð¡2ð+1
[2ð + 1]ð!
â
ð=1
.
Remark 2.27. The previous definitions are q-analogue of classic definitions of
Bernoulli, Euler and Genocchi polynomials. By tending q to 1 from the left side, we
derive to the classic form of these polynomials. That means
limðâ1â
ðžð,ð(ð¥) = ðžð(ð¥), limðâ1â
ðð,ð = ðð, (2.3.6)
limðâ1â
ðºð,ð(ð¥) = ðºð(ð¥), limðâ1â
ðð,ð = ðð. (2.3.7)
In the next chapter, we will introduce some theorems and we reach to a few
properties of these new q-analogues of Bernoulli, Euler and Genocchi polynomials.
limðâ1â
ðµð,ð(ð¥) = ðµð(ð¥), limðâ1â
ðð,ð = ðð, (2.3.8)
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15
Chapter 3
APPROACH TO THE NEW CLASS OF q-BERNOULLI,
q-EULER AND q-GENNOCHI POLYNOMIALS
3.1 Relations to The q-Trigonometric Functions
This chapter is based on [17], we discovered some new results corresponding to the
new definition of q-Bernoulli, q-Euler and q-Genocchi polynomials. The results are
presented one by one as a lemma and propositions. Here, the details of proof and
techniques are given.
Lemma 3.1. Following recurrence formula is satisfied by q-Bernoulli numbers ðð,ð
â[ðð]ð
(â1; ð)ðâð2ðâð
ðð,ð â ðð,ð = {1, ð = 1,0, ð > 1.
ð
ð=0
(3.1.1)
Proof. The statement can be found by the simple multiplication on generating
function of q-Bernoulli numbers (2.3.1). We have
ðµï¿œÌï¿œ(ð¡)ðð(ð¡) = ð¡ + ðµï¿œÌï¿œ(ð¡)
This implies that
ââ[ðð]ð
(â1, ð)ðâð2ðâð
ð
ð=0
â
ð=ð
ðð,ðð¡ð
[ð]ð!= ð¡ +âðð,ð
ð¡ð
[ð]ð!
â
ð=ð
.
Comparing ð¡ð-coefficient observe the expression.
Similar relations are hold for q-Euler and q-Genocchi numbers. If we do the same
thing to their generating functions, we will find the following recurrence formulae
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â[ðð]ð
(â1, ð)ðâð2ðâð
ðð,ð + ðð,ð = {2, ð = 0,0, ð > 0,
ð
ð=0
(3.1.2)
These recurrence formulae help us to evaluate these numbers inductively. The first
few q-Bernoulli, q-Euler, q-Genocchi numbers are given as following. The
interesting thing over here is that, these values coincide with the classic values better
than the previous one. Actually, the odd terms of q-Bernoulli numbers are zero as
classic Bernoulli numbers and lead us to make a connection to a relation with
trigonometric functions.
ð0,ð = 1, ð1,ð = â1
2, ð2,ð =
[3]ð[2]ð â 4
4[3]ð, ð3,ð = 0
ð0,ð = 1, ð1,ð = â1
2, ð2,ð = 0, ð3,ð =
ð(1 + ð)
8,
ð0,ð = 0, ð1,ð = 1, ð2,ð = âð + 1
2, ð3,ð = 0.
Lemma 3.2. The odd coefficients of the q-Bernoulli numbers except the first one are
zero, which means that ðð,ð = 0 for ð = 2ð + 1, ð â â.
Proof. It is the direct consequence of the fact, that the function
ð(ð¡) = âðð,ðð¡ð
[ð]ð!
â
ð=0
â ð1,ðð¡ =ð¡
ðð(ð¡) â 1+ð¡
2=ð¡
2(ðð(ð¡) + 1
ðð(ð¡) â 1) (3.1.4)
is even, and the coefficient of ð¡ð in the McLaurin series of any arbitrary even
function like ð(ð¡), for all odd power ð will be vanished. Itâs based on this fact that if
ð is an even function, then for any ð we have ð(ð)(ð¡) = (â1)ðð(ð)(âð¡) therefore
for any odd ð we lead to ð(ð)(0) = (â1)ðð(ð)(0). Since ðð(âð¡) = (ðð(ð¡))â1
, It
could be happen.
â[ðð]ð
(â1, ð)ðâð2ðâð
ðð,ð + ðð,ð = {2, ð = 1,0, ð > 1.
ð
ð=0
(3.1.3)
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Corollary 3.3. The following identity is true
â [2ðð]ð
(â1, ð)2ðâð22ðâð
ðð,ð = â1
2ðâ2
ð=1
ð > 1 (3.1.5)
Proof. Since ð2ðâ1,ð = 0 for ð > 1, and ð0,ð = 1, simple substitution at recurrence
formula lead us to this expression.
Next lemma shows q-trigonometric functions with q-Bernoulli and q-Genocchiâs
demonstration. We will expand ð¡ððâð(ð¥) and ððð¡ð(ð¥) in terms of q-Genocchi and q-
Bernoulli numbers respectively
Lemma 3.4. The following identities are hold
ð¡ððð¡ð(ð¥) = 1 +âðð,ð(â4)ðð¡2ð
[2ð]ð!
â
ð=1
ð¡ððâð(ð¥)
= ââð2ð+2,ð(2ð¡)2ð+1
[2ð + 2]ð!
â
ð=1
(3.1.6)
Proof. Substitute t by 2it at the generating function of q-Bernoulli number and
remember that ðð(2ðð¡) = ðð(ðð¡) ðžð(ðð¡) = ðð(ðð¡) ( ðð(âðð¡))â1
,
1 â ðð¡ +âðð,ð(2ðð¡)ð
[ð]ð!
â
ð=2
= âðð,ð(2ðð¡)ð
[ð]ð!
â
ð=ð
= ï¿œÌï¿œ(2ðð¡) =2ðð¡
ðð(2ðð¡) â 1
=ð¡ðð(âðð¡)
ð ððð(ð¡)=
ð¡
ð ððð(ð¡)(ððð ð(ð¡) â ðð ððð(ð¡))
= ð¡ððð¡ð(ð¡) â ðð¡
(3.1.7)
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Since ð¡ððð¡ð(ð¡) is even and the odd coefficient of ðð,ð are zero, the first expression is
true. For the next one, putting ð§ = 2ðð¡ at (2.3.4) which is the generating function for
q-Genocchi numbers and we reach to
2ðð¡ +âðð,ð(2ðð¡)ð
[ð]ð!
â
ð=2
= âðð,ð(2ðð¡)ð
[ð]ð!
â
ð=ð
= ï¿œÌï¿œ(2ðð¡) =4ðð¡
ðð(2ðð¡) + 1
=ð¡ðð(âðð¡)
ððð ð(ð¡)
(3.1.8)
In a same way, we reach to ð¡ððð(ð¡) = â ðð,ð(â1)ð(2ð¡)2ðâ1
[2ð]ð!âð=1 . To find the
expression, put ð¥ = ðð¡ instead of ð¥ at ð¡ððð(ð¥). This and writing q-tangent numbers
as the following, together lead us to the interesting identity, which is presented as
follow
ð¡ððâðð¡ = âðð¡ððð(ðð¡) =ðð(ð¡) â ðð(âð¡)
ðð(ð¡) + ðð(âð¡)=ðð(2ð¡) â 1
ðð(2ð¡) + 1= âð2ð+1,ð
(â1)ðð¡2ð+1
[2ð + 1]ð!
â
ð=1
And at the end,
ð2ð+1,ð = ð2ð+2,ð(â1)ðâ122ð+1
[2ð + 2]ð!. (3.1.9)
3.2 Addition and Difference Equations and Corollaries
In this section, by using the q-addition formula, we approach to the new formula for
q-polynomials including q-Bernoulli, q-Euler and q-Genocchi polynomials. This is
the q-analogue of classic expression for these polynomials. In addition, by taking the
ordinary differentiation and q-derivative, we will present some new results as well.
Next lemma presents the q-analogue of additional theorem.
Lemma 3.5. For any complex numbers ð¥, ðŠ, the following statements are true
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19
ðµð,ð(ð¥, ðŠ) = â[ðð]ð
ð
ð=0
ðð,ð(ð¥âšððŠ)ðâð
, ðµð,ð(ð¥, ðŠ)
= â [ðð]ð
ð
ð=0
(â1, ð)ðâð2ðâð
ðµð,ð(ð¥)ðŠðâð
ðžð,ð(ð¥, ðŠ) = â[ðð]ð
ð
ð=0
ðð,ð(ð¥âšððŠ)ðâð
, ðžð,ð(ð¥, ðŠ)
= â[ðð]ð
ð
ð=0
(â1, ð)ðâð2ðâð
ðžð,ð(ð¥)ðŠðâð
ðºð,ð(ð¥, ðŠ) = â[ðð]ð
ð
ð=0
ðð,ð(ð¥âšððŠ)ðâð
, ðºð,ð(ð¥, ðŠ)
= â[ðð]ð
ð
ð=0
(â1, ð)ðâð2ðâð
ðºð,ð(ð¥)ðŠðâð
Proof. The proof is on a base of definition of q-addition, we will do it for q-Bernoulli
polynomials and the remained will be as the same. It is the consequence of the
following identity
âðµð,ð(ð¥, ðŠ)ð¡ð
[ð]ð!
â
ð=0
=ð¡
ðð(ð¡) â 1ðð(ð¡ð¥)ðð(ð¡ðŠ)
= âðð,ðð¡ð
[ð]ð!
â
ð=0
â(ð¥âšððŠ)ðâð ð¡ð
[ð]ð!
â
ð=0
= ââ[ðð]ð
ð
ð=0
ðð,ð(ð¥âšððŠ)ðâð ð¡ð
[ð]ð!.
â
ð=0
(3.2.1)
This is the direct consequence of the definition of q-addition. Actually, q-addition
was defined such that to make q-improved exponential commutative, that means
according to this definition, we can reach to ðð(ð¡ð¥)ðð(ð¡ðŠ) = â (ð¥âšððŠ)ðâð ð¡ð
[ð]ð!âð=0 ,
because the simple calculation for the expansion of ðð(ð¡ð¥) and ðð(ð¡ðŠ) respectively
shows
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20
ðð(ð¡ð¥)ðð(ð¡ðŠ) = (â(â1, ð)ð2ð
(ð¡ð¥)ð
[ð]ð!
â
ð=0
)(â(â1, ð)ð2ð
(ð¡ðŠ)ð
[ð]ð!
â
ð=0
)
= ââ[ðð]ð
(â1, ð)ð(â1, ð)ðâð2ð
ð
ð=0
ð¥ððŠðâðð¡ð
[ð]ð!
â
ð=0
= â(ð¥âšððŠ)ðâð ð¡ð
[ð]ð!
â
ð=0
.
(3.2.2)
Corollary 3.6. For any complex number , we have the following statements
ðµð,ð(ð¥) = â[ðð]ð
(â1, ð)ðâð2ðâð
ð
ð=0
ðð,ðð¥ðâð, ðµð,ð(ð¥, 1) = â [
ðð]ð
ð
ð=0
(â1, ð)ðâð2ðâð
ðµð,ð(ð¥)
ðžð,ð(ð¥) = â[ðð]ð
(â1, ð)ðâð2ðâð
ð
ð=0
ðð,ðð¥ðâð, ðžð,ð(ð¥, 1) = â[
ðð]ð
ð
ð=0
(â1, ð)ðâð2ðâð
ðžð,ð(ð¥)
ðºð,ð(ð¥) = â[ðð]ð
(â1, ð)ðâð2ðâð
ð
ð=0
ðð,ðð¥ðâð, ðºð,ð(ð¥, 1) = â[
ðð]ð
ð
ð=0
(â1, ð)ðâð2ðâð
ðºð,ð(ð¥)
Proof. Itâs easy to substitute y by the suitable values to reach the statements, first
put ðŠ = 0 then put ðŠ = 1, at the previous lemma complete the proof.
Actually, these formulae are the q-analogue of the classic forms, which are
ðµð(ð¥ + 1) = â(ðð)
ð
ð=0
ðµð(ð¥), ðžð(ð¥ + 1) = â(ðð)
ð
ð=0
ðžð(ð¥), ðºð(ð¥ + 1)
= â(ðð)
ð
ð=0
ðºð(ð¥)
Corollary 3.7. q-derivative of q-Bernoulli polynomial is as following
ð·ð (ðµð,ð(ð¥)) = [ð]ððµðâ1,ð(ð¥) + ðµðâ1,ð(ð¥ð)
2 (3.2.3)
Proof. According to the previous corollary, if we know the value of q-Bernoulli
numbers, then we can present q-Bernoulli polynomials in terms of them. Therefore,
by taking q-derivatives from the summation we lead to
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21
ð·ð (ðµð,ð(ð¥)) = ð·ð (â[ðð]ð
(â1, ð)ðâð2ðâð
ð
ð=0
ðð,ðð¥ðâð)
= â[ð]ð!
[ð]ð! [ð â ð â 1]ð!
(â1, ð)ðâð2ðâð
ðâ1
ð=0
ðð,ðð¥ðâðâ1
If we change the order of the summation, we have
ð·ð (ðµð,ð(ð¥)) =[ð]ð
2â
[ð â 1]ð!
[ð]ð! [ð â ð â 1]ð!
(â1, ð)ðâðâ12ðâðâ1
ðâ1
ð=0
(1 + ððâðâ1)ðð,ðð¥ðâðâ1
=[ð]ð
2(â
[ð â 1]ð!
[ð]ð! [ð â ð â 1]ð!
(â1, ð)ðâðâ12ðâðâ1
ðâ1
ð=0
ðð,ðð¥ðâðâ1
+â[ð â 1]ð!
[ð]ð! [ð â ð â 1]ð!
(â1, ð)ðâðâ12ðâðâ1
ðâ1
ð=0
ðð,ð(ðð¥)ðâðâ1)
= [ð]ððµðâ1,ð(ð¥) + ðµðâ1,ð(ð¥ð)
2
In a same way, we can demonstrate the q-derivative of ðžð,ð(ð¥) and ðºð,ð(ð¥) by the
following identities
ð·ð (ðžð,ð(ð¥)) =[ð]ð
2(ðžðâ1,ð(ð¥) + ðžðâ1,ð(ð¥ð)) ,
ð·ð (ðºð,ð(ð¥)) =[ð]ð
2(ðºðâ1,ð(ð¥) + ðºðâ1,ð(ð¥ð)).
(3.2.4)
Next lemma shows another property of these polynomials, which we named them
difference equations.
Lemma 3.8. The following identities are true
ðµð,ð(ð¥, 1) â ðµð,ð(ð¥) =(â1, ð)ðâ12ðâ1
[ð]ðð¥ðâ1 ð ⥠1, (3.2.5)
ðžð,ð(ð¥, 1) â ðžð,ð(ð¥) = 2(â1, ð)ð2ð
[ð]ðð¥ð ð ⥠0, (3.2.6)
ðºð,ð(ð¥, 1) â ðºð,ð(ð¥) = 2(â1, ð)ðâ12ðâ1
[ð]ðð¥ðâ1 ð ⥠1. (3.2.7)
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22
Proof. Since proofs of all statements are similar, we prove it only for q-Bernoulli
difference equation. This can be found from the identity
ð¡ðð(ð¡)
ðð(ð¡) â 1ðð(ð¡ð¥) = ð¡ðð(ð¡ð¥) +
ð¡
ðð(ð¡) â 1ðð(ð¡ð¥) (3.2.8)
It follows that
ââ[ðð]ð
(â1, ð)ðâð2ðâð
ð
ð=0
ðµð,ð(ð¥)ð¡ð
[ð]ð!
â
ð=0
= â(â1, ð)ð2ð
â
ð=0
ð¥ðð¡ð+1
[ð]ð!+âðµð,ð(ð¥)
ð¡ð
[ð]ð!
â
ð=0
Equating the coefficient of ð¡ð completes the proof.
The following familiar expansions will be demonstrated to the means of q-calculus in
the following corollary
ð¥ð =1
ð + 1â(
ð + 1ð
)
ð
ð=0
ðµð(ð¥), (3.2.9)
Corollary 3.9. The following identities hold true
ð¥ð =2ð
[ð]ð(â1, ð)ðâ[
ð + 1ð
]ð
(â1, ð)ð+1âð2ð+1âð
ð
ð=0
ðµð,ð(ð¥), (3.2.12)
ð¥ð =2ðâ1
(â1, ð)ð(â[
ðð]ð
ð
ð=0
(â1, ð)ðâð2ðâð
ðžð,ð(ð¥) + ðžð,ð(ð¥)), (3.2.13)
ð¥ð =1
2(â(
ðð)
ð
ð=0
ðžð(ð¥) + ðžð(ð¥)), (3.2.10)
ð¥ð =1
2(ð + 1)(â(
ð + 1ð
)
ð+1
ð=0
ðºð(ð¥) + ðºð+1(ð¥)). (3.2.11)
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23
ð¥ð =2ðâ1
[ð + 1]ð(â1, ð)ð(â [
ð + 1ð
]ð
(â1, ð)ð+1âð2ð+1âð
ð+1
ð=0
ðºð,ð(ð¥)
+ ðºð+1,ð(ð¥))
(3.2.14)
Proof. Evaluate ð¥ð at the difference equation (3.2.5) and use corollary 3.6 then the
last terms at the summation is vanished and equation is yield.
Lemma 3.10. The following identities hold true
â[ðð]ð
(â1, ð)ðâð2ðâð
ð
ð=0
ðµð,ð(ð¥, ðŠ) â ðµð,ð(ð¥, ðŠ) = [ð]ð(ð¥âšððŠ)ðâ1
, (3.2.15)
Proof. The same technique that we used at (3.8), leads us to these expressions. In
fact, we will use the following identity
ð¡ðð(ð¡)
ðð(ð¡) â 1ðð(ð¡ð¥)ðð(ð¡ðŠ) = ð¡ðð(ð¡ð¥)ðð(ð¡ðŠ) +
ð¡
ðð(ð¡) â 1ðð(ð¡ð¥)ðð(ð¡ðŠ), (3.2.18)
It follows that
ââ[ðð]ð
(â1, ð)ðâð2ðâð
ð
ð=0
ðµð,ð(ð¥, ðŠ)ð¡ð
[ð]ð!
â
ð=0
= â(â1, ð)ð(â1, ð)ðâð
2ð
â
ð=0
ð¥ððŠðâðð¡ð+1
[ð]ð!+âðµð,ð(ð¥, ðŠ)
ð¡ð
[ð]ð!
â
ð=0
,
By equating the coefficient of ð¡ð, we complete the proof.
â[ðð]ð
(â1, ð)ðâð2ðâð
ð
ð=0
ðžð,ð(ð¥, ðŠ) + ðžð,ð(ð¥, ðŠ) = 2(ð¥âšððŠ)ð, (3.2.16)
â[ðð]ð
(â1, ð)ðâð2ðâð
ð
ð=0
ðºð,ð(ð¥, ðŠ) â ðºð,ð(ð¥, ðŠ) = 2[ð]ð(ð¥âšððŠ)ðâ1
. (3.2.17)
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24
3.3 Differential Equations Related to q-Bernoulli Polynomials
The classical Cayley transformation ð§ â ð¶ððŠ(ð§, ð) â1+ðð§
1âðð§, is a good reason to
motivate us to interpret the new formula for ðð(ðð¡). This result leads us to a new
generation of formulae, that we called them q-differential equations for q-Bernoulli
polynomials. Similar results can be done for another q-function. We will start it by
the next proposition.
Proposition 3.11. Assume that ð ⥠1 is a positive integer, then we have
â[ðð]ððð,ðððâð,ð
ð
ð=0
ðð
= âðâ[ðð]ððð,ðððâð,ð
ð
ð=0
[ð â 1]ð âð
2â [
ð â 1ð
]ððð,ðððâðâ1,ð
ðâ1
ð=0
[ð]ð
Proof. By knowing the definition of the improved q-exponential as a production of
some terms, we can write
ðð(ð¡) =â1+ ðð(1 â ð)
ð¡2
1 â ðð(1 â ð)ð¡2
â
ð=0
â ðð(ðð¡) =1 + (1 â ð)
ð¡2
1 â (1 â ð)ð¡2
ðð(ð¡)
= ð¶ððŠ (âð¡
2, 1 â ð) ðð(ð¡)
(3.3.1)
Now use this expression at generating function of q-Bernoulli numbers (2.3.1) and
multiplies it by itself,
ðµï¿œÌï¿œ(ðð¡)ðµï¿œÌï¿œ(ð¡) = (ðµï¿œÌï¿œ(ðð¡) â ððµï¿œÌï¿œ(ð¡) (1 + (1 â ð)ð¡
2))
1
1 â ð
2
ðð(ð¡) + 1 (3.3.2)
The last terms of the above expression can be demonstrated by q-Euler numbers, this
expression lead us to the identity that we assumed at the proposition.
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25
Proposition 3.12. For all ð ⥠1, we have
â[2ðð]ððð,ðð2ðâð,ð
2ð
ð=0
ðð
= âðâ[2ðð]ð
(â1, ð)2ðâð22ðâð
ðð,ð
2ð
ð=0
[ð â 1]ð(â1)ð
+ð(1 â ð)
2â [
2ð â 1ð
]ð
(â1, ð)2ðâðâ122ðâðâ1
ðð,ð[ð â 1]ð(â1)ð
2ðâ1
ð=0
,
â [2ð + 1ð
]ððð,ðð2ðâð+1,ð
2ð+1
ð=0
ðð
= ð â [2ð + 1ð
]ð
(â1, ð)2ðâð+122ðâð+1
ðð,ð
2ð+1
ð=0
[ð â 1]ð(â1)ð
âð(1 â ð)
2â [
2ðð]ð
(â1, ð)2ðâð22ðâð
ðð,ð[ð â 1]ð(â1)ð .
2ð
ð=0
Proof. By knowing quotient rule for q-derivative, take q-derivative from the
generating function (2.3.1), also using (3.3.1) and the fact that ð·ð ( ðð(ð¡)) =
1
2( ðð(ðð¡) + ðð(ð¡)) together, we reach
ðµï¿œÌï¿œ(ðð¡)ðµï¿œÌï¿œ(ð¡) =2 + (1 â ð)ð¡
2 ðð(ð¡)(ð â 1)(ððµï¿œÌï¿œ(ð¡) â ðµï¿œÌï¿œ(ðð¡)) (3.3.3)
Expanding the above expression, and equating ð¡ð-coefficient, leads us to the
proposition. Also we can do the same thing for q-Genocchi and q-Euler generators to
find similar identities.
Proposition 3.13. The following differential equations hold true
ð
ðð¡ðµï¿œÌï¿œ(ð¡) = ðµï¿œÌï¿œ(ð¡) (
1
ð¡â(1 â ð) ðð(ð¡)
ðð(ð¡) â 1(â
4ðð
4 â (1 â ð)2ð2ð
â
ð=0
)) (3.3.4)
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26
Proof. First and second identity can be reached by taking the normal derivatives
respect to t and q respectively. We used product rule temporarily, and demonstrate it
as a summation of these expressions. (3.3.6) is the combination of the (3.3.4) and
(3.3.5).
ð
ðð¡ðµï¿œÌï¿œ(ð¡) â
ð
ðððµï¿œÌï¿œ(ð¡)
=ðµï¿œÌï¿œ(ð¡)
ð¡+ðµï¿œÌï¿œ
2(ð¡) ðð(ð¡)
ð¡(â
4ð¡(ðððâ1 â (ð + 1)ðð) â ðð(1 â ð)
4 â (1 â ð)2ð2ð
â
ð=0
)
(3.3.6)
3.4 Explicit Relationship Between q-Bernoulli and q-Euler
Polynomials
We will study a few numbers of explicit relationships that exist between q-analogues
of two new classes of Euler and Bernoulli polynomials in this section. For this
reason, we will investigate some q-analogues of known results, and some new
formulae and their special cases will be obtained in the following. We demonstrate
some q-extensions of the formulae that are given before at [16].
Theorem 3.14. For any positive integer ð, we have the following relationships
ðµð,ð(ð¥, ðŠ) =1
2â[
ðð]ðððâð (ðµð,ð(ð¥)
ð
ð=0
+â[ðð]ð
ð
ð=0
(â1, ð)ðâð
2ðâððµð,ð(ð¥)
ððâð)ðžðâð,ð(ððŠ)
(3.4.1)
ð
ðððµï¿œÌï¿œ(ð¡) = âðµðÌ
2(ð¡) ðð(ð¡) (â
4ð¡(ðððâ1 â (ð + 1)ðð)
4 â (1 â ð)2ð2ð
â
ð=0
) (3.3.5)
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27
=1
2â [
ðð]ðððâð (ðµð,ð(ð¥) + ðµð,ð (ð¥,
1
ð))
ð
ð=0
ðžðâð,ð(ððŠ).
Proof. To reach (3.4.1), let us start with the following identity
ð¡
ðð(ð¡) â 1ðð(ð¡ð¥)ðð(ð¡ðŠ)
=ð¡
ðð(ð¡) â 1ðð(ð¡ð¥).
ðð (ð¡ð) + 1
2.
2
ðð (ð¡ð) + 1
. ðð (ð¡
ðððŠ)
(3.4.2)
Now expanding the above expression leads us to
âðµð,ð(ð¥, ðŠ)ð¡ð
[ð]ð!
â
ð=0
=1
2âðžð,ð(ððŠ)
ð¡ð
ðð[ð]ð!
â
ð=0
â(â1, ð)ð2ððð
â
ð=0
ð¡ð
[ð]ð!âðµð,ð(ð¥)
ð¡ð
[ð]ð!
â
ð=0
+1
2âðžð,ð(ððŠ)
ð¡ð
ðð[ð]ð!
â
ð=0
âðµð,ð(ð¥)ð¡ð
[ð]ð!
â
ð=0
=: ðŒ1 + ðŒ2
We assumed the first part of addition as ðŒ1 and the second part as ðŒ2. Indeed, for the
second part we have
ðŒ2 =1
2âðžð,ð(ððŠ)
ð¡ð
ðð[ð]ð!
â
ð=0
âðµð,ð(ð¥)ð¡ð
[ð]ð!
â
ð=0
=1
2ââ[
ðð]ðððâððµð,ð(ð¥)ðžðâð,ð(ððŠ)
ð¡ð
[ð]ð!.
ð
ð=0
â
ð=0
On the other hand, the first part or ðŒ1, can be rewritten as
ðŒ1 =1
2âðžð,ð(ððŠ)
ð¡ð
ðð[ð]ð!
â
ð=0
ââ[ðð]ð
(â1, ð)ðâð
2ðâððµð,ð(ð¥)
ð
ð=0
â
ð=0
ð¡ð
ððâð[ð]ð!
=1
2ââ[
ðð]ððžðâð,ð(ððŠ)â[
ðð]ð
(â1, ð)ðâð
2ðâððµð,ð(ð¥)
ððâðððâð
ð
ð=0
ð¡ð
[ð]ð!.
ð
ð=0
â
ð=0
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28
Now if we combine the results at ðŒ1 and ðŒ2 ,we lead to the expression that is equal
to â ðµð,ð(ð¥, ðŠ)ð¡ð
[ð]ð!âð=0 . That means
âðµð,ð(ð¥, ðŠ)ð¡ð
[ð]ð!
â
ð=0
=1
2ââ[
ðð]ðððâð (ðµð,ð(ð¥)
ð
ð=0
â
ð=0
+â[ðð]ð
(â1, ð)ðâð
2ðâððµð,ð(ð¥)
ððâð
ð
ð=0
)ðžðâð,ð(ððŠ)ð¡ð
[ð]ð!.
(3.4.3)
It remains to equating ð¡ð-coefficient to complete the proof.
Corollary 3.15. For any nonnegative integer ð, we have the following statement
ðµð,ð(ð¥, ðŠ) = â[ðð]ð(ðµð,ð(ð¥) +
(â1, ð)ðâ12ðâ1
[ð]ðð¥ðâ1)
ð
ð=0
ðžðâð,ð(ðŠ). (3.4.4)
Proof. This is the special case of previous theorem, where ð = 1. It can be assumed
as a q-analogue of Cheonâs main result [22]
Theorem 3.16. For any positive integer ð, the following relationship between q-
analogue of Bernoulli polynomials and Euler polynomials is true
ðžð,ð(ð¥, ðŠ) =1
[ð + 1]ðâ
1
ðð+1âð[ð + 1ð
]ð(â[
ðð]ð
ð
ð=0
(â1, ð)ðâð
2ðâððžð,ð(ð¥)
ððâð
ð+1
ð=0
â ðžð,ð(ðŠ))ðµð+1âð,ð(ðð¥)
(3.4.5)
Proof. Like the previous theorem, we start by the similar identity. In fact, we can
reach to the proof by using the following identity
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29
2
ðð(ð¡) + 1ðð(ð¡ð¥)ðð(ð¡ðŠ) =
2
ðð(ð¡) + 1ðð(ð¡ðŠ).
ðð (ð¡ð) â 1
ð¡.
ð¡
ðð (ð¡ð) â 1
. ðð (ð¡
ððð¥)
According to the definition of q-Euler polynomials, by substituting and expanding
the terms we have
âðžð,ð(ð¥, ðŠ)ð¡ð
[ð]ð!
â
ð=0
=âðžð,ð(ðŠ)ð¡ð
[ð]ð!
â
ð=0
â(â1, ð)ððð2ð
â
ð=0
ð¡ðâ1
[ð]ð!âðµð,ð(ðð¥)
ð¡ð
ðð[ð]ð!
â
ð=0
ââðžð,ð(ðŠ)ð¡ðâ1
[ð]ð!
â
ð=0
âðµð,ð(ðð¥)ð¡ð
ðð[ð]ð!
â
ð=0
=: ðŒ1 â ðŒ2
Indeed,
ðŒ2 =1
ð¡âðžð,ð(ðŠ)
ð¡ð
[ð]ð!
â
ð=0
âðµð,ð(ðð¥)ð¡ð
ðð[ð]ð!
â
ð=0
=1
ð¡ââ[
ðð]ð
1
ððâððžð,ð(ðŠ)ðµðâð,ð(ðð¥)
ð¡ð
[ð]ð!
ð
ð=0
â
ð=0
= â1
[ð + 1]ðâ[
ð + 1ð
]ð
1
ðð+1âððžð,ð(ðŠ)ðµð+1âð,ð(ðð¥)
ð¡ð
[ð]ð!.
ð+1
ð=0
â
ð=0
In addition for ðŒ1, we have
ðŒ1 =1
ð¡âðµð,ð(ðð¥)
ð¡ð
ðð[ð]ð!
â
ð=0
ââ[ðð]ð
(â1, ð)ðâð2ðâð
ðžð,ð(ðŠ)
ð
ð=0
â
ð=0
ð¡ð
ððâð[ð]ð!
=1
ð¡ââ[
ðð]ððµðâð,ð(ðð¥)â[
ðð]ð
(â1, ð)ðâð
2ðâððžð,ð(ðŠ)
ððâðððâð
ð
ð=0
ð¡ð
[ð]ð!
ð
ð=0
â
ð=0
=â1
[ð + 1]ðâ[
ð + 1ð]ð
ðµð+1âð,ð(ðð¥)â [ðð]ð
(â1, ð)ðâð
2ðâððžð,ð(ðŠ)
ðð+1âð
ð
ð=0
ð¡ð
[ð]ð!
ð+1
ð=0
â
ð=0
.
Combining ðŒ1 and ðŒ2, and equating ð¡ð- coefficient together complete the proof.
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30
Theorem 3.17. For any nonnegative integer ð, the relationship between q-analogue
of Bernoulli polynomials and Genocchi polynomials can be described as following
ðºð,ð(ð¥, ðŠ) =1
[ð + 1]ðâ
1
ððâð[ð + 1ð
]ð(â[
ðð]ð
ð
ð=0
(â1, ð)ðâð
2ðâððºð,ð(ð¥)
ððâð
ð+1
ð=0
â ðºð,ð(ð¥))ðµð+1âð,ð(ððŠ)
(3.4.6)
ðµð,ð(ð¥, ðŠ) =1
2[ð + 1]ðâ
1
ððâð[ð + 1ð
]ð(â[
ðð]ð
ð
ð=0
(â1, ð)ðâð
2ðâððµð,ð(ð¥)
ððâð
ð+1
ð=0
+ ðµð,ð(ð¥))ðºð+1âð,ð(ððŠ)
(3.4.7)
Proof. At this theorem, we use the same technique to find a relationship between q-
analogue of Genocchi and Bernoulli polynomials, which is completely new.
Proof is straightforward. Like the previous one, first assume the following identity,
2ð¡
ðð(ð¡) + 1ðð(ð¡ð¥)ðð(ð¡ðŠ)
=2ð¡
ðð(ð¡) + 1ðð(ð¡ð¥). (ðð (
ð¡
ð) â 1)
ð
ð¡.
ð¡ð
ðð (ð¡ð) â 1
. ðð (ð¡
ðððŠ)
Now, substitute the generating function (2.3.4). A simple calculation shows that
âðºð,ð(ð¥, ðŠ)ð¡ð
[ð]ð!
â
ð=0
=ð
ð¡âðºð,ð(ð¥)
ð¡ð
[ð]ð!
â
ð=0
â(â1, ð)ððð2ð
â
ð=0
ð¡ð
[ð]ð!âðµð,ð(ððŠ)
ð¡ð
ðð[ð]ð!
â
ð=0
âð
ð¡âðºð,ð(ð¥)
ð¡ð
[ð]ð!
â
ð=0
âðµð,ð(ððŠ)ð¡ð
ðð[ð]ð!
â
ð=0
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31
By using the Cauchy product of two series, it can be rewritten
âðºð,ð(ð¥, ðŠ)ð¡ð
[ð]ð!
â
ð=0
=ð
ð¡â(â[
ðð]ð
(â1, ð)ðâðððâð2ðâð
ðºð,ð(ðŠ)âðºð,ð(ð¥)
ð
ð=0
)
â
ð=0
ð¡ð
[ð]ð!âðµð,ð(ððŠ)
ð¡ð
ðð[ð]ð!
â
ð=0
=ð
ð¡ââ
1
ððâð[ðð]ð
ð
ð=0
(â[ðð]ð
(â1, ð)ðâð
ððâð2ðâððºð,ð(ðŠ)âðºð,ð(ð¥)
ð
ð=0
)
â
ð=0
ðµðâð,ð(ððŠ)ð¡ð
[ð]ð!
= â1
[ð + 1]ðâ
1
ððâð[ð + 1ð
]ð(â[
ðð]ð
(â1, ð)ðâð
ððâð2ðâððºð,ð(ðŠ)âðºð,ð(ð¥)
ð
ð=0
)ðµð+1âð,ð(ððŠ)ð¡ð
[ð]ð!
ð+1
ð=0
â
ð=0
.
Now equating ð¡ð- coefficient to find the first identity. The second one can be proved
in a same way.
We will finish the chapter by focusing on the symmetric properties of the given
polynomials. In fact, part (b) at theorem (2.17) and definition of ð-polynomials by
the generating function (2.3.1) leads us to the following proposition
Proposition 3.18. ð-Bernoulli, ð-Euler and ð-Gennoci numbers has the following
property
ðâ(ð2)ðð,ð = ðð,ðâ1 , ð
â(ð2)ðð,ð = ðð,ðâ1 ððð ð
â(ð2)ðð,ð = ðð,ðâ1 (3.4.8)
Proof. The proof is based on the fact that [ð]ðâ1! = ðâ(ð2)[ð]ð!. Since ðð(ð§) =
ð1ð
(ð§), we have
ð¡
ðð(ð¡) â 1= âðð,ð
ð¡ð
[ð]ð!
â
ð=0
=ð¡
ððâ1(ð¡) â 1= âðð,ðâ1
ð¡ð
[ð]ðâ1!
â
ð=0
(3.4.9)
Equating ð¡ð-coefficient, leads us to (3.4.1).
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32
Remark 3.19. The previous proposition, gives us a tool to evaluate the
corresponding values of ðð,ðâ1 by knowingðð,ð.
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33
Chapter 4
UNIFICATION OF q-EXPONENTIAL FUNCTION AND
RELATED POLYNOMIALS
4.1 Preliminary Results
We will define the new class of q-exponential function in this section. In fact, we
will add a parameter to the old definition. In this way, we reach to the unification of
q-exponential function and by changing this parameter; we lead to the different kind
of the q-exponential functions that defined before. Moreover, this parameter helps us
to lead to a group of new q-exponential functions as well. We will study the
important properties of q-exponential function, by taking some restrictions on this
parameter.
Definition 4.1. Let ðŒ(ð, ð) be a function of ð and ð, such that ðŒ(ð, ð) â 1, where ð
tends to one from the left side. We define new general q-exponential function as
following
ðð,ðŒ(ð§) = âð§ð
[ð]ð!ðŒ(ð, ð)
â
ð=0
(4.1.1)
In the special case where ðŒ(ð, ð) = 1, ðŒ(ð, ð) = ð(ð2) and ðŒ(ð, ð) =
(â1,ð)ð
2ð we reach
to ðð(ð§), ðžð(ð§) and ðð(ð§) respectively.
At the next lemma, we will discuss about the conditions that make ðð,ðŒ(ð§)
convergent. There are some restrictions, which have to be considered. Since ðð,ðŒ(ð§)
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34
is the q-analogue of exponential function, ðŒ(ð, ð) approaches to 1, where q tends one
from the left side.
Lemma 4.2. If limðââ |ðŒ(ð,ð+1)
[ð+1]ð!ðŒ(ð,ð)| does exist and is equal to ð ,then ðð,ðŒ(ð§) as a q-
exponential function is analytic in the region |ð§| < ðâ1.
Proof. Radius of convergence can be obtained by computing the following limit
limðââ
|ð§ð+1ðŒ(ð, ð + 1)
[ð + 1]ð!| |
[ð]ð!
ð§ððŒ(ð, ð)| = lim
ðââ|
ðŒ(ð, ð + 1)
[ð + 1]ð! ðŒ(ð, ð)| |ð§| (4.1.2)
Then, for ð â 1 we can use d'Alembert's test and we find the radius of convergence.
Example 4.3. Let ðŒ(ð, ð) = 1, ðŒ(ð, ð) = ð(ð2) and ðŒ(ð, ð) =
(â1,ð)ð
2ð, then we reach
to, ðð(ð§), ðžð(ð§) and improved q-exponential function ðð(ð§) respectively. Then the
radius of convergence becomes 1
|1âð|, infinity and
2
|1âð| respectively where 0 < |ð| <
1.
Now, with this q-exponential function, we define the new class of q-Bernoulli
numbers and polynomials. Next definition denotes a general class of these new q-
numbers and polynomials.
Definition 4.4. Assume that ð is a complex number such that 0 < |ð| < 1. Then we
can define q-analogue of the following functions in the meaning of generating
function including Bernoulli numbers ðð,ð,ðŒ and polynomials ðµð,ð,ðŒ(ð¥, ðŠ) and Euler
numbers ðð,ð,ðŒ and polynomials ðžð,ð,ðŒ(ð¥, ðŠ) and the Genocchi numbers ðð,ð,ðŒ and
polynomials ðºð,ð,ðŒ(ð¥, ðŠ) in two variables ð¥, ðŠ respectively
ðµï¿œÌï¿œ â
ð¡
ðð,ðŒ(ð¡) â 1= âðð,ð,ðŒ
ð¡ð
[ð]ð!
â
ð=0
, |ð¡| < 2ð, (4.1.3)
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35
ð¡
ðð,ðŒ(ð¡) â 1 ðð,ðŒ(ð¡ð¥) ðð,ðŒ(ð¡ðŠ) = âðµð,ð,ðŒ(ð¥, ðŠ)
ð¡ð
[ð]ð!
â
ð=0
, |ð¡| < 2ð, (4.1.4)
ðžï¿œÌï¿œ â
2
ðð,ðŒ(ð¡) + 1= âðð,ð,ðŒ
ð¡ð
[ð]ð!
â
ð=0
, |ð¡| < ð, (4.1.5)
2
ðð,ðŒ(ð¡) + 1 ðð,ðŒ(ð¡ð¥) ðð,ðŒ(ð¡ðŠ) = âðžð,ð,ðŒ(ð¥, ðŠ)
ð¡ð
[ð]ð!
â
ð=0
, |ð¡| < 2ð, (4.1.6)
ðºï¿œÌï¿œ â
2ð¡
ðð,ðŒ(ð¡) + 1= âðð,ð,ðŒ
ð¡ð
[ð]ð!
â
ð=0
, |ð¡| < ð, (4.1.7)
2ð¡
ðð,ðŒ(ð¡) + 1 ðð,ðŒ(ð¡ð¥) ðð,ðŒ(ð¡ðŠ) = âðºð,ð,ðŒ(ð¥, ðŠ)
ð¡ð
[ð]ð!
â
ð=0
, |ð¡| < ð. (4.1.8)
If the convergence conditions are hold for q-exponential function, it is obvious that
by tending q to 1 from the left side, we lead to the classic definition of these
polynomials. We mention that ðŒ(ð, ð) is respect to q and n. In addition by tending q
to 1â», ðð,ðŒ(ð§) approach to the ordinary exponential function. That means
ðð,ð,ðŒ = ðµð,ð,ðŒ(0), limðâ1â
ðµð,ð,ðŒ(ð¥, ðŠ) = ðµð(ð¥, ðŠ), limðâ1â
ðð,ð,ðŒ = ðð, (4.1.9)
ðð,ð,ðŒ = ðžð,ð,ðŒ(0), limðâ1â
ðžð,ð,ðŒ(ð¥, ðŠ) = ðžð(ð¥, ðŠ), limðâ1â
ðð,ð,ðŒ = ðð, (4.1.10)
ðð,ð,ðŒ = ðºð,ð,ðŒ(0), limðâ1â
ðºð,ð,ðŒ(ð¥, ðŠ) = ðºð(ð¥, ðŠ), limðâ1â
ðð,ð,ðŒ = ðð. (4.1.11)
Our purpose in this chapter is presenting a few results and relations for the newly
defined q-Bernoulli and q-Euler polynomials. In the next section we will discuss
about some restriction for ðŒ(ð, ð), such that the familar results discovered. we will
focus on two main properties of q-exponential function, first in which situation
ðð,ðŒ(ð§) = ððâ1,ðŒ(ð§), second we investigate the conditions for ðŒ(ð, ð) such that
ðð,ðŒ(âð§) = (ðð,ðŒ(ð§))â1
. A lot of classical results are found by these two
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36
properties.The form of new type of q-exponential function, motivate us to define a
new q-addition and q-subtraction like a Daehee formula as follow
(ð¥âšððŠ)ðââ(
ðð)ð
ð
ð=0
ðŒ(ð, ð â ð)ðŒ(ð, ð)ð¥ððŠðâð, ð = 0,1,2, ⊠(4.1.12)
(ð¥ âð ðŠ)ðââ(
ðð)ð
ð
ð=0
ðŒ(ð, ð â ð)ðŒ(ð, ð)ð¥ð(âðŠ)ðâð, ð = 0,1,2, ⊠(4.1.13)
4.2 New Exponential Function and Its Properties
We shall provide some conditions on ðŒ(ð, ð) to reach two main properties that
discussed before. First we try to find out, in which situation ðð,ðŒ(ð§) = ððâ1,ðŒ(ð§).
Following lemma is related to this property.
Lemma 4.5. The new q-exponential function ðð,ðŒ(ð§), satisfy ðð,ðŒ(ð§) = ððâ1,ðŒ(ð§), if
and only if ð(ð2)ðŒ(ðâ1, ð) = ðŒ(ð, ð).
Proof. The proof is based on the fact that [ð]ðâ1! = ðâ(ð2)[ð]ð!, therefore
ððâ1,ðŒ(ðâ1)(ð§) = âð§ð
[ð]ðâ1!ðŒ(ðâ1, ð)
â
ð=0
= âð§ð
[ð]ð!ðŒ(ð, ð)
â
ð=0
= ðð,ðŒ(ð§) (4.2.1)
Corollary 4.6. If ðŒ(ð, ð) is in a form of polynomial that means ðŒ(ð, ð) = â ððððð
ð=0 ,
to satisfy ðð,ðŒ(ð§) = ððâ1,ðŒ(ð§), we have
ððð(ðŒ(ð, ð)) = ð = (ð2) â ð †(
ð2) ,
ðð+ð = ðð+ð ððð ð = 0,1, . . . , ð â ð
(4.2.2)
Where ð is the leading index, such that ðð â 0 and for 0 †ð < ð, ðð = 0.
Proof. First, we want to mention that â ðððð=0 = 1, becuase ðŒ(ð, ð) approches to 1,
where q tends one from the left side. In addition as we assumed ðŒ(ð, ð) = â ðððð=0 ðð ,
by substituting qâ»Â¹ instead of q we have
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37
ð(ð2)ðŒ(ðâ1, ð) = ð
(ð2)âð
âðð
ð
ð=0
ðð = ðŒ(ð, ð) =âðð
ð
ð=0
ðð (4.2.3)
Now equate the coefficient of ðð , to reach the statement.
Example 4.7. Simplest example of the previous corollary will be happened when
(ð, ð) = ð(ð2)
2 . This case leads us to the following exponential function
ðð,ðŒ(ð§) = âð§ð
[ð]ð!ð(ð2)
2
â
ð=0
& ððâ1,ðŒ(ðâ1)(ð§) = ðð,ðŒ(ð§) (4.2.4)
Another example will be occurred if ðŒ(ð, ð) =(â1,ð)ð
2ð=
(1+ð)(1+ð2)âŠ(1+ðð)
2ðâ1. Now use
q-binomial formula (2.8) to reach ðŒ(ð, ð) =1
2ðâ (
ðð)ððð(ðâ1)
2ðð=0 . As we expect,
where q tends 1 from the left side, ðŒ(ð, ð) approach to 1. This presentation is not in a
form of previous corollary, however ð(ð2)ðŒ(ðâ1, ð) = ðŒ(ð, ð). This parameter leads
us to the improved q-exponential function as following
ðð(ð§) = ðð,ðŒ(ð§) = âð§ð
[ð]ð!
(â1, ð)ð2ð
â
ð=0
& ððâ1(ð§) = ðð(ð§) (4.2.5)
Some properties of q-Bernoulli polynomials that are corresponding to this improved
q-exponential function were studied at previous chapter.
Remark 4.8. It's obvious that if we substitute q to qâ»Â¹, in any kind of q-exponential
function and derive to another q-analogue of exponential function, the parameter
ðŒ(ð, ð) will change to ðœ(ð, ð), and ð(ð2)ðŒ(ðâ1, ð) = ðœ(ð, ð). The famous case is
standard q-exponential function
ððâ1(ð§) = ððâ1,ðŒ(ðâ1)(ð§) = âð§ð
[ð]ðâ1!
â
ð=0
= âð§ð
[ð]ð!
â
ð=0
ð(ð2)= ðžð(ð§) (4.2.6)
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38
ð(ð2)ðŒ(ðâ1, ð) = ð
(ð2)= ðœ(ð, ð) (4.2.7)
Proposition 4.9. The general q-exponential function ðð,ðŒ(ð§) satisfy ðð,ðŒ(âð§) =
(ðð,ðŒ(ð§))â1
, if and only if
2â(
ðð)ð
ðâ1
ð=0
(â1)ððŒ(ð, ð â ð)ðŒ(ð, ð) = (ðð)
ð(â1)ð+1ðŒ2(ð, ð)&
ðŒ(ð, 0) = ±1 ð€âððð ð = 2ð ððð ð = 1,2, . ..
(4.2.8)
Proof. This condition can be rewritten as (1 âð 1)ð= 0 for any ð â â. Since
ðð,ðŒ(âð§)ðð,ðŒ(ð§) = 1 has to be hold, we write the expansion for this equation, then
ðð,ðŒ(âð§)ðð,ðŒ(ð§) = â(â(ðð)ð(â1)ððŒ(ð, ð â ð)ðŒ(ð, ð)
ð
ð=0
)
â
ð=0
= 1 (4.2.9)
Let call the expression on a bracket as ðœð,ð. If n is an odd number, then
ðœðâð,ð = (ð â ðð
)ð(â1)ðâððŒ(ð, ð â ð)ðŒ(ð, ð)
= (ðð)ð(â1)ððŒ(ð, ð â ð)ðŒ(ð, ð)
= â ðœð,ð ð€âððð ð = 0,1, . . . , ð
(4.2.10)
Therefore, for n as an odd number, we have the trivial equation. Since (ð â ðð
)ð=
(ðð)ðThe same discussion for even n and equating zâ¿-coefficient together lead us to
the proof.
Remark 4.10. The previous proposition can be rewritten as a system of nonlinear
equations. For instance, let write ðŒ(ð, ð) as ðŒð, then the following system shows a
condition for ðŒð. We mention that ðŒð = 1 where ð â 1â and ðŒâ = ±1.
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39
{
2ðŒâðŒâ â (
21)ððŒâðŒâ = 0
2ðŒâðŒâ â 2 (41)ððŒâðŒâ + (
42)ððŒâðŒâ = 0
2ðŒ6ðŒ1 â 2(61)ððŒ5ðŒ2 + 2(
62)ððŒ4ðŒ3 â 2(
63)ððŒ3ðŒ3 = 0
...
2ðŒððŒ1 â 2(ð1)ððŒðâ1ðŒ2 + 2(
ð â 22
)ððŒðâ2ðŒ3 ââ¯+ (â1)
ð2â (
ðð2â)ð
ðŒð2âðŒð
2â= 0
For even n, we have ð 2â equations and ð unknown variables. In this case we can find
ðŒð respect to ð 2â parameters by the recurence formula. For example, some few terms
can be found as follow
{
ðŒâ = ±1
ðŒâ =1 + ð
2
1
ðŒ1
ðŒ4 =[4]ð
2ðŒ12([2]ððŒ3 â
[3]ð!
4ðŒ1)
ðŒ6 = (61)ð+ (
63)ðâ1
2((62)ð(1 + ð
2
1
ðŒ1) (
[4]ð
2ðŒ12([2]ððŒ3 â
[3]ð!
4ðŒ1)))
The familiar solution of this system is ðŒ(ð, ð) =(â1,ð)ð
2ð. This ðŒ(ð, ð) leads us to the
improved exponential function. On the other hand, we can assume that all ðŒð for odd
k are 1. Then by solving the system for these parameters, we reach to another
exponential function that satisfies ðð,ðŒ(âð§) = (ðð,ðŒ(ð§))â1
.
In the next lemma, we will discuss about the q-derivative of this q-exponential
function. Here, we assume the special case, which covers well known q-exponential
functions.
Lemma 4.11. If ðŒ(ð,ð+1)
ðŒ(ð,ð) can be demonstrated as a polynomial of ð, that means
ðŒ(ð,ð+1)
ðŒ(ð,ð) = â ðð
ðð=0 ðð, then ð·ð ( ðð,ðŒ(ð§)) = â ðð
ðð=0 ðð,ðŒ (ð§ð
ð
ð).
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Proof. We can prove it by using the following identity
ð·ð ( ðð,ðŒ(ð§)) = âð§ðâ1
[ð â 1]ð!
â
ð=1
ðŒ(ð, ð) = âð§ð
[ð]ð!
â
ð=0
ðŒ(ð, ð) (âðð
ð
ð=0
ðð)
= âðð
ð
ð=0
â(ð§ð
ðð)
ð
[ð]ð!
â
ð=0
ðŒ(ð, ð) = âðð
ð
ð=0
ðð,ðŒ (ð§ððð) .
(4.2.11)
Example 4.12. For ðŒ(ð, ð) = 1, ðŒ(ð, ð) = ð(ð2) and ðŒ(ð, ð) =
(â1,ð)ð
2ð, the ratio of
ðŒ(ð,ð+1)
ðŒ(ð,ð) becomes 1, ðâ¿ and ((1 + ðâ¿)/2) respectively. Therefore the following
derivatives hold true
ð·ð ( ðð(ð§)) = ðð(ð§) & ð·ð ( ðžð(ð§)) = ðžð(ð§ð) & ð·ð ( ðð(ð§)) = ðð(ð§) + ðð(ð§ð)
2
4.3 Related q-Bernoulli Polynomial
In this section, we will study the related q-Bernoulli polynomials, q-Euler
polynomials and q-Genocchi polynomials. The discussion of properties of general q-
exponential at the previous section, give us the proper tools to reach to the general
properties of these polynomials related to ðŒ(ð, ð).
Lemma 4.13. The condition ðð,ðŒ(âð§) = (ðð,ðŒ(ð§))â1
and ðŒ(ð, 1) = 1 together
provides that the odd coefficient of related q-Bernoulli numbers except the first one
becomes zero. That means ðð,ð,ðŒ = 0 where ð = 2ð + 1, ð â â.
Proof. The proof is similar to (3.2) and based on the fact that the following function
is even. I mention that the condition ðŒ(ð, 1) = 1 and (4.2.8) together imply
that ð1,ð,ðŒ = â1
2.
ð(ð¡) = âðð,ð,ðŒð¡ð
[ð]ð!
â
ð=0
â ð1,ð,ðŒð¡ =ð¡
ðð,ðŒ(ð¡) â 1+ð¡
2=ð¡
2(ðð,ðŒ(ð¡) + 1
ðð,ðŒ(ð¡) â 1) (4.3.1)
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Lemma 4.14. If ðŒ(ð, ð) as a parameter of ðð,ðŒ(ð§) satisfy ðŒ(ð,ð+1)
ðŒ(ð,ð) = â ðð
ðð=0 ðð,
then we have
ð·ð (ðµð,ð,ðŒ(ð¥)) = [ð]ðâðð
ð
ð=0
ðµðâ1,ð,ðŒ (ð¥ððð) (4.3.2)
Proof. Use lemma 4.11 similar corollary 3.7, we reach to the relation. Moreover in a
same way for q-Euler and q-Genocchi polynomials we have
ð·ð (ðžð,ð,ðŒ(ð¥)) = [ð]ðâðð
ð
ð=0
ðžðâ1,ð,ðŒ (ð¥ððð) (4.3.3)
ð·ð (ðºð,ð,ðŒ(ð¥)) = [ð]ðâðð
ð
ð=0
ðºðâ1,ð,ðŒ (ð¥ððð). (4.3.4)
4.4 Unification of q-Numbers
In this section, we study some properties of related q-numbers including q-Bernoulli,
q-Euler and q-Gennochi numbers. For this reason we investigate these numbers that
is generated by the unified q-exponential numbers. We reach to the general case of
these numbers. In addition, any new definition of these q-numbers can be
demonstrate in this form and we can study the general case of them by applying that
two properties of q-exponential function which is discussed in the previous sections.
As I mention it before, all the lemma and propositions at the previous chapter can be
interpreted in a new way. The proofs and techniques are as the same. Only some
format of the polynomials will be changed. For example the symmetric proposition
can be rewritten as following
Proposition 4.15. If the symmetric condition is hold, that means ðð,ðŒ(ð§) = ððâ1,ðŒ(ð§),
then ð-Bernoulli, ð-Euler and ð-Gennoci numbers has the following property
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ðâ(ð2)ðð,ð,ðŒ = ðð,ðâ1,ðŒ , ð
â(ð2)ðð,ð,ðŒ = ðð,ðâ1,ðŒ ððð ð
â(ð2)ðð,ð,ðŒ = ðð,ðâ1,ðŒ (4.4.1)
Since the proof is similar, it is not given. The symmetric conditions for ðð,ðŒ(ð§) can
be hold from 4.5 to 4.8.
Another interesting property of these numbers can be found in the next lemma. We
can demonstrate all q-Bernoulli numbers by knowing their values where ð belongs to
the interval of [0, 1]. When ð is a real number, we can assume this interval as a space
of the range of the probability random variable.
Lemma 4.16. If the symmetric condition is hold, that means ðð,ðŒ(ð§) = ððâ1,ðŒ(ð§),
then related q-Bernoulli, q-Euler and q-Gennochi numbers can be found if we have
the value of them for 0 < |ð| < 1.
Proof. If 0 < |ð| < 1, then by the assumption of lemma, the value of ðð,ð,ðŒ, ðð,ð,ðŒ
and ðð,ð,ðŒ are known. If ð is outside of this region, ðâ1 will be inside of this region
and by using the previous proposition we can compute the value of these numbers.
In the middle of the last century, q-analogues of the Bernoulli numbers were
introduced by Carlitz [4]. The recurrence relation that he used for these numbers was
as following
â(ðð)
ð
ð=0
ðµððð+1 â ðµð = {
1, ð = 1,0, ð > 1.
(4.4.2)
In the next proposition we will give the recurrence formulae for the related q-
Bernoulli numbers and another q-numbers as well. These presentations are based on
the generating function of these numbers by using the unification of q-exponential
function.
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Proposition 4.17. Related q-Bernoulli numbers, q-Euler numbers and q-Gennochi
numbers to the new q-exponential function can be evaluated by the following
recurrence formulae
â(ðð)ððŒ(ð, ð â ð)ðð,ð,ðŒ â ðð,ð,ðŒ = {
1, ð = 10, ð > 1,
ð
ð=0
(4.4.3)
â(ðð)ððŒ(ð, ð â ð)ðð,ð,ðŒ + ðð,ð,ðŒ = {
2, ð = 0,0, ð > 0,
ð
ð=0
(4.4.4)
â(ðð)ððŒ(ð, ð â ð)ðð,ð,ðŒ + ðð,ð,ðŒ = {
2, ð = 1,0, ð > 1.
ð
ð=0
(4.4.5)
Proof. Related q-numbers are defined by the means of the generating function at
(4.1.3), (4.1.5) and (4.1.7) respectively. If we multiply each side by the
corresponding terms and using Cauchy product of the series, we reach to these
recurrence relations.
Remark 4.18. The previous proposition can be used to evaluate these numbers one
by one. In this case, by assuming that ðŒ0 = ðŒ(ð, 0) = 1, few numbers of related q-
Bernoulli, q-Euler and q-Gennochi numbers can be obtained as following
ð0,ð,ðŒ =1
ðŒ1, ð1,ð,ðŒ = â
ðŒ2(ðŒ1)2[2]ð
, ð2,ð,ðŒ =1
[3]ððŒ1(â
ðŒ3ðŒ1+(ðŒ2)
2[3]ð(ðŒ1)2[2]ð
),
ð0,ð,ðŒ = 1, ð1,ð,ðŒ = âðŒ12, ð2,ð,ðŒ =
1
2((ðŒ2)
2[2]ð
2â ðŒ2),
ð0,ð,ðŒ = 0, ð1,ð,ðŒ = 1, ð2,ð,ðŒ = âð + 1
2ðŒ1.
If we continue to evaluate the sequence of q-Bernoulli numbers, we can see that the
lemma 4.13 hold true. That means the odd coefficients of the general q-Bernoulli
numbers are zero, in a condition that the related q-exponential function has the
symmetric property. In addition, by knowing the recurrence relation of the q-
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Bernoulli numbers we can reach to the corresponding q-exponential function. For
instance, the q-Bernoulli numbers that were introduced by Carlitz can be traced in a
same way. By comparing the terms we can see that
ðŒ(ð, ð â ð) =[ð â ð]ð!
(ð â ð)!
[ð]ð!
ð!
ð!
[ð]ð!ðð+1 (4.4.6)
Now, we can also find the corresponding q-exponential function such that this
relation is hold. This form of unification leads us to a lot of new q-exponential
function with interesting properties.
This unification of q-exponential function gives us a tool to redefine the generating
functions according to this q-exponential function. In addition we can change the
form of generating function to lead a general class of these polynomials as well.
For instance, we can assume the following definition.
Definition 4.19. For the polynomial of a degree m ðð(ð¡), we can define the sequence
of numbers {ðŸð} by the means of generating function as follows
ðð(ð¡)
ðð,ðŒ(ð¡) ± 1= âðŸð
â
ð=0
ð¡ð
[ð]ð!. (4.4.7)
In the special case, when the degree of ðð(ð¡) is one and simply is equal to t, 2 and 2t
we can reach to the q-Bernoulli, q-Euler and q-Gennochi numbers respectively. In a
case that the degree of ðð(ð¡) is higher than one, these numbers can be demonstrated
as a combination of these numbers.
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