Q-Genocchi Numbers and Polynomials Associated with Fermionic P-Adic Invariant Integrals on Zp

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2008, Article ID 232187, 8 pages doi:10.1155/2008/232187

Research Article q-Genocchi Numbers and Polynomials Associated with Fermionic p-Adic Invariant Integrals on Zp

Leechae Jang1 and Taekyun Kim2

1 Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, South Korea 2 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea

Correspondence should be addressed to Leechae Jang, [email protected]

Received 3 April 2008; Accepted 22 April 2008

Recommended by Ferhan Atici

The main purpose of this paper is to present a systemic study of some families of multiple Genocchi numbers and polynomials. In particular, by using the fermionic p-adic invariant integral on Zp, p we construct -adic Genocchi numbers and polynomials of higher order. Finally, we derive the Gk x kk nk ∞ − ll xn following interesting formula: nk,q 2 ! k l0 d0d1···dk k−1,di∈N 1 , where Gk x q k nk,q are the -Genocchi polynomials of order .

Copyright q 2008 L. Jang and T. Kim. 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.

1. Introduction

Let p be a fixed odd prime number. Throughout this paper, Zp, Qp, C, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field and the p-adic completion of the algebraic closure of Qp.Letνp be the normalized ν p exponential valuation of Cp with |p|p p p 1/p. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C,orap-adic number q ∈ Cp. If q ∈ C, one normally assumes |q| < 1. If q ∈ Cp, then we assume that |1 − q|p < 1, see 1–6. In C, the ordinary Euler polynomials are defined as ∞ tn 2 ext E x , |t| <π. et n n 1.1 1 n0 ! x ,E E δ In the case 0 n 0 n are called Euler numbers, see 1–13 .Let 0,n be the Kronecker symbol. From 1.1 we derive the following relation: E , E n E δ ,n∈ N, 0 1 1 n 2 0,n 1.2 2 Abstract and Applied Analysis

n cf. 7–13. Here, we use the technique method notation by replacing E by En n ≥ 0, , − / , , / ,..., E k , ,.... symbolically. The ﬁrst few are 1 1 2 0 1 4 and 2k 0for 1 2 A sequence consisting of the Genocchi numbers Gn satisﬁes the following relations:

t ∞ tn 2 G , |t| <π, et n n 1.3 1 n0 ! G ,G G G ··· G ,k , , ,..., see 11, 12 .Itsatisfies 1 1 3 5 7 2k1 0 1 2 3 and even coefficients are given by G − 2n B nE , 2n 2 1 2 2n 2 2n−1 0 1.4 where Bn is Bernoulli numbers. The first few Genocchi numbers for even integers are −1, 1, −3, 17, −155, 2073,....The first few prime Genocchi numbers are −3 and 17, which occur at n 6 and 8. There are no others with n<105. We now define the Genocchi polynomials as follows: t ∞ tn 2 ext G x , |t| <π. et n n 1.5 1 n0 !

Thus, we note that n n G x G xn−l. n l l 1.6 l0

x In this paper, we use the following notations: xq 1 − q /1 − q and x−q 1 x −q /1 q.LetUDZp be the space of uniformly diﬀerentiable functions on Zp.Forf ∈ UDZp,thep-adic q-deformed fermionic integral on Zp is deﬁned as

N p−1 1 x I−qf fxdμ−qx lim fx−q , 1.7 N→∞ pN Zp q x0 see 1–4. The fermionic p-adic invariant integral on Zp can be obtained as q → 1. That is,

I− flim I−qf fxdμ− x. 1.8 1 q→ 1 1 Zp

From 1.8, we easily derive the following integral equation related to fermionic invariant p- adic integral on Zp: I f I f f , −1 1 −1 2 0 1.9

f xfx where 1 1 , see 5 . The purpose of this paper is to present a systemic study of some families of multiple Genocchi numbers and polynomials by using the fermionic multivariate p-adic invariant integral on Zp. In addition, we will investigate some interesting identities related to Genocchi numbers and polynomials. L. Jang and T. Kim 3

2. Genocchi numbers associated with fermionic p-adic invariant integral on Zp

From 1.9 we can derive t ∞ tn t extdμ x 2 G , −1 et n n Zp 1 n0 ! 2.1 t ∞ tn t exytdμ y 2 ext G x , −1 et n n Zp 1 n0 ! where Gnx are Genocchi polynomials. It is easy to check that t ∞ n n tn t e2x1tdμ x 2 t t 1 lG . −1 t −t sech 2 l 2.2 Z e e l n p 2 n0 l0 !

By comparing the coeﬃcientonbothsidesin2.1, we easily see that G x x yndμ x n1 . −1 n 2.3 Zp 1

Therefore, we obtain the following proposition.

Proposition 2.1. For k ∈ Z,

n x y dμ− xGn x/n i Zp 1 1 1 (Witt’s formula for Genocchi polynomials); x t n1 n l n e 2 1 dμ− x /n / 1 Gl nn − ···n − l ii Zp 1 1 1 1 2 l0 l 2 ,where l 1 1/l!. ϑ {x ∈ C ||x| ≤ } C i − 1/2 ∈ ϑ Let Cp p p 1 be the integer ring of p.Wenotethat 1 Cp .By using Taylor expansion, we see that

∞ xn ∞ − nx2n ∞ − nx2n1 eix in 1 i 1 . n n n 2.4 n0 ! n0 2 ! n0 2 1 !

In the p-adic number ﬁeld, sin x and cos x are deﬁned as

x x3 x5 x7 sin x − − ··· , 1! 3! 5! 7! 2.5 x2 x4 x6 cos x 1 − − ··· . 2! 4! 6! From 2.4 and 2.5,wederive

ix e cos x i sin x. 2.6

This is equivalent to eix e−ix eix − e−ix cos x , sin x . 2.7 2 2i 4 Abstract and Applied Analysis

By 2.7, we easily see that ∞ intn t 2 e2x1itdμ x x ndμ x . sec eit e−it −1 2 1 −1 n 2.8 Zp n0 Zp !

2n1 ﬃ x dμ− x n ∈ Z N ∪{ } It is not di cult to show that Zp 2 1 1 0for 0 .From 2.8 ,we note that ∞ tn ∞ t2n t in x ndμ x − n x 2ndμ x . sec 2 1 −1 n 1 2 1 −1 n 2.9 n0 Zp ! n0 Zp 2 !

Thus, we have ∞ 2n1 n t2n1 t t 1 − n 2 1 lG . sec 1 l 2 l n 2.10 2 n0 l0 2 1 !

Now we consider the fermionic multivariate p-adic invariant integral on Zp as follows: k ∞ n k x x ···x t k t k t t ··· e 1 2 k dμ x ···dμ x G , −1 1 −1 k 2 t t n 2.11 Z Z e ··· e n! p p 1 1 n0 k-times k-times

k where Gn are the nth Genocchi number of order k. By comparing the coeﬃcient on both sides k k k G G ··· Gn in 2.11 , we see that 0 1 0, and ··· x x ··· x ndμ x ···dμ x n k Gk , 1 2 k −1 1 −1 k k nk 2.12 Z Z p p k-times where n kk is the Jordan factor which is deﬁned by n kk n k ···n 1. Thus, we note that Gk ··· x x ··· x ndμ x ···dμ x nk , 1 2 k −1 1 −1 k nk n 2.13 Zp Zp n ! k-times for k ∈ N,n∈ Z.

Theorem 2.2. For n ∈ Z,k∈ N, Gk ··· x x ··· x ndμ x ···dμ x nk . 1 2 k −1 1 −1 k nk n 2.14 Zp Zp n ! k-times L. Jang and T. Kim 5

The multinomial coeﬃcient is well known as n n l l lk x x ··· xk x 1 x 2 ···x . 1 2 l ,...,l 1 2 k 2.15 1 k l1···lkn l ,...,l > 1 k 0

Therefore, we obtain the following corollary.

Corollary 2.3. For n ∈ Z, k ∈ N,

k G n Gl Gl Gl nk 1 1 2 1 ··· k 1 . l ,...,l l l l nk 2.16 1 k 1 1 2 1 k 1 n n! l1···lkn l ,...,l > 1 k 0

For q ∈ Cp with |1 − q| < 1, it is not diﬃcult to show that t t qxextdμ x 2 . −1 qet 2.17 Zp 1

Now, we deﬁne the q-extension of the Genocchi numbers as follows:

t ∞ tn 2 G . qet n,q n 2.18 1 n0 !

By 2.17 and 2.18, we easily see that G n1,q qxxndμ x. n −1 2.19 1 Zp

With the same motivation to construct the Genocchi polynomials of higher order, we can consider the q-extension of higher-order Genocchi numbers as follows: k x x ···k− x xx x ···x t t ··· q 2 2 3 1 k e 1 2 k dμ x ···dμ x −1 1 −1 k Z Z p p k-times k k ∞ n 2.20 t 2 xt k t e Gn,q , et qet ··· qk−1et n! 1 1 1 n0 k-times

k where Gn,q are the q-Genocchi polynomials of order k.Thebasicq-natural numbers are deﬁned as − qn 1 2 n−1 nq 1 q q ··· q ,n∈ N. 2.21 1 − q

The q-factorial of n is deﬁned as n−1 nq! nqn − 1q ···2q1q 1 q ··· q ··· 1 q · 1. 2.22 6 Abstract and Applied Analysis

The q-binomial coefficient is also defined as n nq! nqn − 1q ···n − k 1q . k n − k k k 2.23 q q! q! q! n n nn − ···n − k /k . q ffi Note that limq→1 k q k 1 1 ! The -binomial coe cient satisfies the following recurrsion formula: n n n n n 1 qk qn−k . k k − k k − k 2.24 q 1 q q 1 q q From this recurrsion formula, we can derive n ·d ·d ···k·d q0 0 1 1 k . k 2.25 q d0d1···dkk−1 di∈N The q-binomial expansion is given by n n n k a bqi−1 q an−kbk, k 2 i1 k0 q 2.26 n ∞ − n k − − bqi−1 1 1 bk. 1 k i1 k0 q By 2.20 and 2.26, we see that k x x ···k− x xx x ···x t t ··· q 2 2 3 1 k e 1 2 k dμ x ···dμ x −1 1 −1 k Z Z p p k-times k k k i− t −1 xt t 2 1 − − q 1 e e i1 2.27 ∞ k l − k k 1 l lxt t 2 −1 e l l0 q ∞ ∞ k l − tn tk k 1 − ll xn . 2 l 1 n n0 l0 q ! Therefore, we obtain the following theorem.

Theorem 2.4. For n ∈ Z,k∈ N,wehave x x ···k− x n ··· q 2 2 3 1 k x x ··· x dμ x ···dμ x 1 2 k −1 1 −1 k Z Z p p k-times 2.28 ∞ k l − k 1 − ll xn. 2 l 1 l0 q L. Jang and T. Kim 7

By 2.20,itisnotdiﬃcult to show that k n k x x ···k− x k G xk ··· q 2 2 3 1 k × x x x ··· x nk,q ! 1 2 k k Z Z p p k-times 2.29 × dμ x ···dμ x n k , −1 1 −1 k k k k k G G ··· Gn,q , 0,q 1,q 0 where n 0, 1, 2,....Therefore, we obtain the following corollary.

Corollary 2.5. For n ∈ Z,k∈ N,

k G x ∞ nk,q k k l − 1 l n 2 −1 l x . 2.30 k nk l ! k l0 q

Corollary 2.6. For n ∈ Z,k∈ N, ∞ k k n k ·d ·d ···k·d l n G x k q0 0 1 1 k − l x . nk,q 2 ! k 1 2.31 l0 d0d1···dkk−1 di∈N

Acknowledgment

The present research has been conducted by the research grant of Kwangwoon University in 2008.

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