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Abstract and Applied Analysis
Volume 2008, Article ID 898471, 7 pages
http://dx.doi.org/10.1155/2008/898471
Research Article

On Genocchi Numbers and Polynomials

Department of Mathematics, Kyungpook National University, Tagegu 702-701, South Korea

Received 10 May 2008; Accepted 21 July 2008

Academic Editor: Lance Littlejohn

Copyright © 2008 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.

Abstract

The main purpose of this paper is to study the distribution of Genocchi polynomials. Finally, we construct the Genocchi zeta function which interpolates Genocchi polynomials at negative integers.

1. Introduction

Let be a fixed odd prime number. Throughout this paper, , and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of the algebraic closure of . Let be the normalized exponential valuation of with . When one talks about -extension, is variously considered as an indeterminate, a complex, , or a -adic number, . If one normally assumes . If , then we assume . The ordinary Genocchi polynomials are defined as the generating function: For a fixed positive integer with , set(cf. [130]), where satisfies the condition . We say that is uniformly differential function at and write if the difference quotients, , have a limit as . Throughout this paper, we use the following notation:For , the fermionic -adic invariant -integral on is defined assee [127]. Note thatIn this paper, we investigate some interesting integral equations related to . From these integral equations related to , we can derive many interesting properties of Genocchi numbers and polynomials. The main purpose of this paper is to derive the distribution relations of the Genocchi polynomials, and to construct the Genocchi zeta function which interpolates the Genocchi polynomials at negative integers.

2. Genocchi Numbers and Polynomials

The Genocchi numbers are defined aswhere is replaced by , symbolically. The Genocchi polynomials are also defined asFrom (2.1), we note that , and . The fermionic -adic invariant integral on is defined asLet be translation with . Then we have the following integral equation. Note that From (2.3), we can deriveThus, we obtainFor , we haveBy (2.6) and (2.7), if we take , we easily see thatThus, we haveIf , then we know thatThus, we getLet be the Dirichlet character with conductor , with . Then, we consider the generalized Genocchi numbers attached to as follows:where . From (2.7) and (2.12), we note thatBy (2.12) and (2.13), it is not difficult to show thatBy (2.6) and (2.15), we obtain the following theorem.

Theorem 2.1. Let with , and let be the Dirichlet character with conductor . Then, one has

3. Genocchi Zeta Function

Let be the generating function of in complex plane as follows:Then, we show thatBy (3.1) and (3.2), we easily see thatTherefore, we obtain the following proposition.

Proposition 3.1. For , one has

From Proposition 3.1, we can derive the Genocchi zeta function which interpolates Genocchi polynomials at negative integers.

For , we define the Hurwitz-type Genocchi zeta function as follows.

Definition 3.2. For ,

By Proposition 3.1 and Definition 3.2, we obtain the following theorem.

Theorem 3.3. For , one has

Let be the Dirichlet character with conductor , with , and let be the generating function in of . Then, we haveFrom (3.7), we deriveBy (3.1), (3.2), and (3.8), we easily see thatFrom (3.9), we can deriveThus, we haveNow, we consider the Dirichlet-type Genocchi -function in complex plane as follows. For , defineBy (3.11) and (3.12), we obtain the following theorem.

Theorem 3.4. Let be the Dirichlet character with conductor , with , and let . Then, one has

Remark 3.5. In [1], we can observe the value of Genocchi zeta function at positive integers as follows:

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