Báo cáo hóa học: " Research Article Multivariate Twisted p-Adic q-Integral on Zp Associated with Twisted q-Bernoulli Polynomials and Numbers" pdf

Recently, many authors have studied twisted q-Bernoulli polynomials by using the p-adic invariant q-integral onZp.. In this paper, we define the twisted p-adic q-integral onZpand extend

Volume 2010, Article ID 579509, 6 pages

doi:10.1155/2010/579509

Research Article

and Numbers

Seog-Hoon Rim, Eun-Jung Moon, Sun-Jung Lee,

and Jeong-Hee Jin

Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

Correspondence should be addressed to Seog-Hoon Rim,shrim@knu.ac.kr

Received 19 June 2010; Accepted 2 October 2010

Academic Editor: Ulrich Abel

Copyrightq 2010 Seog-Hoon Rim et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Recently, many authors have studied twisted q-Bernoulli polynomials by using the p-adic invariant

q-integral onZp In this paper, we define the twisted p-adic q-integral onZpand extend our result

to the twisted q-Bernoulli polynomials and numbers Finally, we derive some various identities related to the twisted q-Bernoulli polynomials.

1 Introduction

Let p be a fixed prime number Throughout this paper, the symbols Z, Z p , Q p , C, and C pwill

denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational

numbers, the complex number field, and the completion of the algebraic closure of Qp ,

respectively LetN be the set of natural numbers and Z  N ∪ {0} Let v pbe the normalized exponential valuation ofCpwith|p| p  p −v p p  1/p.

When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ C, or p-adic number q ∈ C p If q ∈ C, one normally assumes that |q| < 1 If

q ∈ C p , then we assume that |q − 1| p < 1.

For n ∈ N, let T p be the p-adic locally constant space defined by

n≥1

C p n  lim

where C p n  {ζ ∈ C p | ζ p n

 1 for some n ≥ 0} is the cyclic group of order p n

Let UDZ p be the space of uniformly differentiable function on Zp

For f ∈ UDZ p , the p-adic invariant q-integral on Z pis defined as

I q



f





Zp

f xdμ q x  lim

N → ∞

1



p N

q

pN−1

x0

f xq x , 1.2

compare with1 3

It is well known that the twisted q-Bernoulli polynomials of order k are defined as

e xt

t

e t ζq − 1

k

∞

n0

β n,ζ,q k x t n

and β k

n,ζ,q  β k

n,ζ,q 0 are called the twisted q-Bernoulli numbers of order k When k  1, the polynomials and numbers are called the twisted q-Bernoulli polynomials and numbers, respectively When k  1 and q  1, the polynomials and numbers are called the twisted Bernoulli polynomials and numbers, respectively When k  1, q  1, and ζ  1, the

polynomials and numbers are called the ordinary Bernoulli polynomials and numbers, respectively

Many authors have studied the twisted q-Bernoulli polynomials by using the properties of the p-adic invariant q-integral on Z p cf 4 In this paper, we define the

twisted p-adic q-integral on Z p and extend our result to the twisted q-Bernoulli polynomials and numbers Finally, we derive some various identities related to the twisted q-Bernoulli

polynomials

2 Multivariate Twisted p-Adic q-Integral on Zp Associated with

Twisted q-Bernoulli Polynomials

In this section, we assume that q ∈ C pwith|q−1| p < 1 For ζ ∈ T p, we define theq, ζ-numbers

as

k q,ζ 1− q k ζ

1− q , for k ∈ Z p 2.1

Note thatk q  k q,1  1 − q k /1 − q.

Let us define

n

k q,ζ k n q,ζ!

q,ζ!n − kq,ζ!, 2.2

wherek q,ζ! k q,ζ k − 1 q,ζ· · · 1q,ζ Note thatn

k  n

k1,1  n!/k!n − k!.

Now we construct the twisted p-adic q-integral on Z pas follows:

I q,ζ



f





Zp

f xdμ q,ζ x

 lim

N → ∞

pN−1

x0

f xμ q,ζ

x  p NZp

 lim

N → ∞

1



p N

q

pN−1

x0

f xq x ζ x ,

2.3

where μ q,ζ x  p NZp   q x ζ x /p Nq From the definition of the twisted p-adic q-integral on

Zp , we can consider the twisted q-Bernoulli polynomials and numbers of order k as follows:

β n,q,ζ k x 



Zk x1 x2 · · ·  x k  x n

q dμ q,ζ x1dμ q,ζ x2 · · · dμ q,ζ x k

 lim

N → ∞

1



p Nk

q

pN−1

x1, ,x k0

x1 x2 · · ·  x k  x n

q q x1x2···x k ζ x1x2···x k

  1

1− qn

n



l0

n

l −1l

q lx lim

N → ∞

1



p Nk

q

pN−1

x1, ,x k0

q l1x1···l1x k ζ x1···x k

  1

1− qn

n



l0

n

l −1l

q lx l  1 k

l  1 k q,ζ

.

2.4

In the special case x  0 in 2.4, β n,q,ζ k 0  β k n,q,ζ are called the twisted q-Bernoulli numbers of order k.

If we take k  1 and ζ  1 in 2.4, we can easily see that

β n,q x   1

1− qn

n



l0

n

l −1l

q lx l  1 l  1 q . 2.5

compare with4

Theorem 2.1 For k ∈ Zand ζ ∈ T p , we have

β k n,q,ζ x   1

1− qn

n



l0

n

l −1l

q lx l  1 k

l  1 k q,ζ

Moreover, if we take x  0 in Theorem2.1, then we have the following identity for the

twisted q-Bernoull numbers

β n,q,ζ k   1

1− qn

n



l0

n

l −1l l  1 k

l  1 k q,ζ

From the definition of multivariate twisted p-adic q-integral, we also see that

β k n,q,ζ x 



Zk x1 x2 · · ·  x k  x n

q dμ q,ζ x1dμ q,ζ x2 · · · dμ q,ζ x k

n

l0

n

lx x n−l q



Zk x1 x2 · · ·  x kl

q dμ q,ζ x1dμ q,ζ x2 · · · dμ q,ζ x k

n

l0

n

lx x n−l

q β k l,q,ζ

2.8

Corollary 2.2 For k ∈ Zand ζ ∈ T p , one obtains

β n,q,ζ k x n

l0

n

lx x n−l

Note that

q nx1···x kn

l0

n

l x1 · · ·  x kl

We have



Zk q nx1···x kdμ q,ζ x1dμ q,ζ x2 · · · dμ q,ζ x k n

l0

n

l

β k l,q,ζ 2.11

It is easy to see that



Zk q nx1···x kdμ q,ζ x1dμ q,ζ x2 · · · dμ q,ζ x k

 lim

N → ∞

1



p Nk q

pN−1

x1, ,x k0

q nx1···x kq x1···x k ζ x1···x k  n  1 k

n  1 k q,ζ

.

2.12

By2.11 and 2.12, we obtain the following theorem

Theorem 2.3 For n ∈ Z, k ∈ N and ζ ∈ Tp , one has

n



l0

n

l

β k l,q,ζ n  1 k

n  1 k q,ζ

Now we consider the modified extension of the twisted q-Bernoulli polynomials of order k as follows:

B k n,q,ζ x   1

1− qn

n



i0 −1i

n

ix



Zk q k l1 k−lix i dμ q,ζ x1 · · · dμ q,ζ x k . 2.14

In the special case x  0, we write B n,q,ζ k  B n,q,ζ k 0, which are called the modified extension of the twisted q-Bernoulli numbers of order k.

From2.14, we derive that

B k n,q,ζ x   1

1− qn

n



i0

−1i

n i

i  k · · · i  1

i  k q,ζ · · · i  1 q,ζ

q ix

  1

1− qn

n



i0

−1i

n i

ik

k



k!

ik

k



q,ζ k q,ζ!q ix

2.15

Therefore, we obtain the following theorem

Theorem 2.4 For n ∈ Z, k ∈ N and ζ ∈ Tp , one has

B k n,q,ζ x   1

1− qn

n



i0

−1i

n i

ik

k



k!

ik

k



q,ζ k q,ζ!q ix 2.16

Now, we define B −k n,q,ζ x as follows:

B n,q,ζ −k x   1

1− qn

n



i0

−1in

i q ix



Zk q k l1 k−lix i dμ q,ζ x1 · · · dμ q,ζ x k. 2.17

By2.17, we can see that

B −k n,q,ζ x   1

1− qn

n



i0

−1i

n i

ik k



q,ζ k q,ζ!

ik

k



Therefore, we obtain the following theorem

Theorem 2.5 For n ∈ Z, k ∈ N and ζ ∈ Tp , one has

B n,q,ζ −k x   1

1− qn

n



i0

−1i

i  k

nk

n−i



k q,ζ!

nk

k



ix 2.19

In2.19, we can see the relations between the binomial coefficients and the modified

extension of the twisted q-Bernoulli polynomials of order k.

Acknowledgments

The authors would like to thank the anonymous referee for his/her excellent detail comments and suggestions This Research was supported by Kyungpook National University Research Fund, 2010

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2002

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