art. generalizations of poly-bernoulli numbers and polynomials.pdf

8/11/2019 art. Generalizations of poly-Bernoulli numbers and polynomials.pdf

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arXiv:1212

.3989v1

[math.NT

]17Dec2012

IJMC: INTERNATIONAL JOURNAL OF MATHEMATICAL COMBINATORICS (2010), Vol 2 #A07-14

GENERALIZATIONS OF POLY-BERNOULLI NUMBERS AND

POLYNOMIALS

Hassan Jolany and M.R.Darafsheh, R.Eizadi Alikelaye

School of Mathematics,college of science,university of Tehran, Tehran, Iran

[email protected]

Received April 12, 2010: , Accepted: May 28, 2010

Keywords:Poly-Bernoulli polynomials,Euler polynomials

,generalized Euler polynomials,generalized poly-Bernoulli polynomials

Abstract

The Concepts of poly-Bernoulli numbers B(k)n , poly-Bernoulli polynomials Bkn(t) and the

generalized poly-bernoulli numbers B(k)n (a, b) are generalized to B

(k)n (t,a,b,c) which is called

the generalized poly-Bernoulli polynomials depending on real parameters a,b,c .Some prop-erties of these polynomials and some relationships between Bkn , B

(k)n (t) , B

(k)n (a, b) and

B(k)n (t,a,b,c) are established.

1. Introduction

In this paper we shall develop a number of generalizations of the poly-Bernoulli numbersand polynomials , and obtain some results about these generalizations.They are fundamentalobjects in the theory of special functions.

Euler numbers are denoted with Bk and are the coefficients of Taylor expansion of thefunction t

et1 ,as following;

tet 1

=

k=0

Bk tk

k!.

The Euler polynomials En(x) are expressed in the following series

2ext

et + 1=

k=0

Ek(x)tk

k!.

for more details, see [1]-[4].
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In [10], Q.M.Luo, F.Oi and L.Debnath defined the generalization of Euler polynomialsEk(x,a,b,c) which are expressed in the following series :

2cxt

bt +at =

k=0

Ek(x,a,b,c) tk

k!.

where a,b,c Z+.They proved that

I) for a= 1 and b= c = e

Ek(x+ 1) =k

j=0

k

j Ej(x), (1)and

Ek(x+ 1) +Ek(x) = 2xk, (2)

II) for a= 1 and b= c ,

Ek(x+ 1, 1, b , b) +Ek(x, 1, b , b) = 2xk(ln b)k. (3)

In[5], Kaneko introduced and studied poly-Bernoulli numbers which generalize the clas-

sical Bernoulli numbers. Poly-Bernoulli numbers B(k)n with k Zand n N, appear in the

following power series:

Lik(1 ex)

1 ex =

n=0

B(k)ntn

n!, ()

where k Z and

Lik(z) =

m=1

zm

mk. |z|


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et 1

et 1

t0

1

et 1

t0

... 1

et 1

t0

t

et 1dtdt...dt

=n=0

B(k)ntn

n!.

In the special case, one can see B(1)n =Bn.

Definition 1 The poly-Bernoulli polynomials,B(k)n (t), are appeared in the expansion of

Lik(1ex)

1ex ext

as following,

Lik(1 ex

)1 ex ext =

n=0

B

(k)

n (t)n! xn (4)

for more details, see [6] [11].

Proposition 1 (Kaneko theorem[6]) The Poly-Bernoulli numbers of non-negative index k,satisfy the following

B(k)n = (1)n

n+1

m=1

(1)m1(m 1)!

n

m 1

mk

, (5)

and for negative index k, we have

B(k)n =

min(n,k)j=0

(j!)2

n+ 1j+ 1

k+ 1j+ 1

, (6)

where

nm= (1)

m

m!

m

l=0

(1)l ml ln m, n 0 (7)

Definition 2 Let a, b > 0 and a = b. The generalized poly-Bernoulli numbers B(k)n (a, b),

the generalized poly-Bernoulli polynomialsB(k)n (t, a, b) and the polynomialB

(k)n (t,a,b,c) are

appeared in the following series respectively;

Lik(1 (ab)t)

bt at =

n=0

B(k)n (a, b)

n! tn |t| a >0, we have

B(k)n (x+y, a, b , c) =l=0

n

l

(ln c)nlB

(k)l (x; a,b,c)y

nl =n

l=0

n

l

(ln c)nlB

(k)l (y, a, b , c)x

nl.

(17)

Proof: We have

Lik(1 (ab)t)

bt at c(x+y)t =

n=0

B(k)n (x+y; a, b , c)tn

n!


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=Lik(1 (ab)

t)

bt at cxt.cyt =

n=0B(k)n (x; a,b,c)

tn

n!

i=0yi(ln c)i

i! ti

=n=0

nl=0

n

l

ynl(ln c)nlB

(k)l (x,a,b,c)

tn

n!

so we getLik(1 (ab)t)

bt at c(x+y)t =

Lik(1 (ab)t

bt at cytcxt

=n=0

nl=0

n

l

xnl(ln c)nlB

(k)l (y, a, b , c)

tn

n!.

Theorem 3 Letx R andn 0.For every positive real numbers a,b and c such thata =bandb > a >0, we have

B(k)n (x; a, b , c) =n

l=0

n

l

(ln c)nlB

(k)l

ln b

ln a+ ln b

(ln a+ ln b)lxnl, (18)

B(k)n (x; a,b,c) =n

l=0

l

j=0

(1)lj nl l

j (ln c)nl(ln b)lj(ln a+ ln b)jB(k)j xnk. (19)

Proof:

(18)By applying Theorems 1 and 2 we know ,

B(k)n (x; a, b , c) =n

l=0

n

l

(ln c)nlB

(k)l (a, b)x

nl

and

B(k)n (a, b) =B (k)n ( lnb

ln a+ ln b)(ln a+ ln b)n

The relation (18)will follow if we combine these formulae.

(19) proof is similar to(18).

Now,we give some results about derivatives and integrals of the generalized poly-Bernoullipolynomials in the following theorem.


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Theorem 4 Letx R.Ifa, bandc >0 , a =b andb > a > 0 ,For any non-negative integerl and real numbers andwe have

lB(k)n (x,a,b,c)

xl =

n!

(n l)! (lnc

)

lB(k)

nl(x,a,b,c

) (20)

B(k)n (x,a,b,c)dx= 1

(n+ 1) ln c[B

(k)n+1( , a, b , c) B

(k)n+1( , a, b , c)] (21)

Proof: By applying (20) and (21) can be proved by induction on n.

In [9], GI-Sang Cheon investigated the classical relationship between Bernoulli and Eulerpolynomials, in this paper we study the relationship between the generalized poly-Bernoulli

and Euler polynomials.

Theorem 5 Forb >0 we have

B(k1)n (x+y, 1, b , b) =1

2

nk=0

n

k

[B(k1)n (y, 1, b , b) +B

(k1)n (y+ 1, 1, b , b)]Enk(x, 1, b , b).

Proof:We know

B(k1)n (x+y, 1, b , b) =

k=0

nk (ln b)nkB(k1)k (y; 1, b , b)xnk

andEk(x+y, 1, b , b) +Ek(x, 1, b , b) = 2x

k(ln b)k

So, We obtain B(k1)n (x+y, 1, b , b) =

1

2

n

k=0

n

k

(ln b)nkB

(k1)k (y; 1, b , b)

1

(ln b)nk(Enk(x, 1, b , b) +Enk(x+ 1, 1, b , b))

=1

2

nk=0

n

k

B

(k1)k (y; 1, b , b)

Enk(x, 1, b , b) +

nkj=0

n kj

Ej(x, 1, b , b)

=1

2

nk=0

n

k

B

(k1)k (y; 1, b , b)Enk(x, 1, b , b)


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+1

2

nj=0

n

j

Ej(x; 1, b , b)

njk=0

n jk

B

(k1)k (y, 1, b , b)

=

1

2

nk=0

nk

B(k1)

k (y; 1, b , b)Enk(x, 1, b , b)+

1

2

nj=0

n

j

B

(k1)nj(y+ 1; 1, b , b)Ej(x, 1, b , b)

So we have

B(k1)n (x+y, 1, b , b) =1

2

nk=0

n

k

[B(k1)n (y, 1, b , b) +B

(k1)n (y+ 1, 1, b , b)]Enk(x, 1, b , b).

Corollary 1 In theorem 5, if we setk1= 1 andb= e, we obtain

Bn(x) =n

(k=0),(k=1)

n

k

BkEnk(x).

For more detail see [7].

Acknowledgement:The authors would like to thank the referee for has valuable com-

ments concerning theorem 5.

References

[1] T.Arakawa and M.Kaneko:On poly-Bernoulli numbers,Comment.Math.univ.st.Pauli48,(1999),159-167

[2] B.N. Oue and F.Qi:Generalization of Bernoulli polynomials ,internat.j.math.ed.sci.tech33, (2002),No,3,428-431

[3] M.-S.Kim and T.Kim:An explicit formula on the generalized Bernoulli number withorder n,Indian.j.pure and applied math 31, (2000),1455-1466

[4] Hassan Jolany and M.R.Darafsheh:Some another remarks on the generalization ofBernoulli and Euler polynomials,Scientia magna Vol5,NO 3

[5] M.Kaneko:poly-Bernoulli numbers ,journal de theorides numbers de bordeaux. 9,(1997),221-228


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[6] Y.Hamahata,H.Masubuch:Special Multi-Poly- Bernoulli numbers,Journal of Integer se-quences vol 10, (2007)

[7] H.M.Srivastava and A.Pinter:Remarks on some relationships Between the Bernoulli and

Euler polynomials,Applied .Math.Letter 17, (2004)375-380

[8] Chad Brewbaker :A combinatorial Interpretion of the poly-Bernoulli numbers and twofermat analogues, Integers journal 8, (2008)

[9] GI-Sang Cheon:A note on the Bernoulli and Euler polynomials,Applied.Math.Letter16, (2003),365-368

[10] Q.M.Luo,F.Oi and L.Debnath Generalization of Euler numbers and polynomi-als,Int.J.Math.Sci (2003)3893-3901

[11] Y.Hamahata,H.Masubuchi Recurrence formulae for Multi-poly-Bernoulli num-bers,Integers journal 7(2007)

[12] Hassan Jolany, Explicit formula for generalization of poly-Bernoulli numbers and poly-nomials with a,b,c parameters.http://arxiv.org/pdf/1109.1387v1.pdf
http://arxiv.org/pdf/1109.1387v1.pdfhttp://arxiv.org/pdf/1109.1387v1.pdf

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