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

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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.
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