Abstract and Applied Analysis

Volume 2012 (2012), Article ID 679495, 9 pages

doi:10.1155/2012/679495

Research Article

On the Barnes' Type Related to Multiple Genocchi Polynomials on

J. Y. Kang, H. Y. Lee, N. S. Jung, and C. S. Ryoo

Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 29 July 2012; Revised 17 August 2012; Accepted 21 August 2012

Academic Editor: Józef Banaś

Copyright © 2012 J. Y. Kang 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.

Using fermionic -adic invariant integral on , we construct the Barnes' type multiple Genocchi numbers and polynomials. From those numbers and polynomials, we derive the twisted Barnes' type multiple Genocchi numbers and polynomials. Moreover, we will find the Barnes' type multiple Genocchi zeta function.

1. Introduction

Recently, theoretical physicists have devised ultrametric structures similar to tree-like structures arising in the study of physical systems because the fact is that the physical space may no longer be Archimedean seems plausible to some mathematical physicists at a very small distance. They have looked for construct-related models using -adic numbers and -adic analysis. So, -adic numbers are used in mathematical physics (in particular string theory, field theory) as well as in other areas of natural sciences which complicated fractal behavior and hierarchical structures. Also, -Volkenborn integral which is made by Kim is used in the functional equation of the -zeta function, the -Stirling numbers, and Mahler theory of integration with respect to the ring together with Iwasawas -adic - function. So we study the -Gnocchi numbers and polynomials in -type of special generating functions (see [1, 2]).

Let be a fixed odd prime number. Throughout this paper, , and 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 , respectively. Let be the set of natural numbers and . Let be the normalized exponential valuation of with (see[1–15]).

Let be a fixed integer and let be a fixed prime number. For any positive integer , we set

where lies in .

We say that is uniformly differentiable function at a point , and we write if the difference quotients such that

have a limit as .

As well-known definition, the Genocchi polynomials are defined by

with the usual convention of replacing by . In the special case, , are called the th Genocchi numbers.

For , Kim defined the -deformed fermionic -adic integral on ,

Note that

If we take , by (1.4), we see that

By the same method, we note that

And note that

Barnes' type multiple zeta function depends on the parameters that will be assumed to be positive. It is defined by

for , . Barnes showed that had a meromorphic continuation in s (with simple pole only at ) and defined his multiple gamma function in terms of the -derivative at , which may be recalled here as follows:

2. Barnes' Type Multiple Genocchi Polynomials

In this section, we assume that . For , , we define Barnes' type multiple Genocchi polynomials as follows:

In the special case, , are called the th Barnes' type multiple Genocchi numbers.

Thus, we have

For , , we can readily prove the following (2.3) from (2.2):

where is a binomial coefficient.

Theorem 2.1 (Property of distribution of ). For , and with ,

Let be Dirichlet's character with conductor with :

By (2.1) and (2.5), we see that

From (2.2) and (2.6), we have

By (2.5), we see that

We define the generalized Barnes' type multiple Genocchi polynomials attached to as follows: for and ,

For simple calculation of fermionic -adic invariant on , we note the following theorem.

Theorem 2.2 (Property of distribution of ).

Theorem 2.2 is distribution relation for the generalized Barnaes' type multiple Genocchi polynomials attached to . In the special case, , are called the generalized Barnes' type multiple Genocchi numbers attached to .

By (2.8) and (2.9), we get

3. Twisted Barnes' Type Multiple Genocchi Polynomials

Let , where is the cyclic group of order . For , we denote by the locally constant function . If we take , then we get

By (3.1), we easily see that

From (3.1) and (3.2), we define the twisted Genocchi numbers and the generalized twisted Genocchi numbers attached to as follows:

In (3.1), (3.2), and (3.3), we get

By using fermionic multivariate -adic invariant integral on , we define the twisted Barnes' type multiple Genocchi polynomials as follows:

are called the twisted Barnes' type multiple Genocchi polynomials. In the special case, , are called the twisted Barnes' type multiple Genocchi numbers.

From (3.5), we can derive the following:

Let be the primitive Drichlet character with conductor with . Then we define the generalized twisted Barnes' type multiple Genocchi numbers attached to :

From (3.7), we see that

By using (3.7), we easily see that

Theorem 3.1 (Property of distribution of ).

4. Remark

Let be taken nonnegative in complex plane. We consider the Barnes' type multiple Genocchi zeta function as follows:

where , and .

By (4.1) and (2.7), we get

where .

Define

where , and .

From (4.3), we note that

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