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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 679495, 9 pages
On the Barnes' Type Related to Multiple Genocchi Polynomials on
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.
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.
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
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 .
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 ).
Let be taken nonnegative in complex plane. We consider the Barnes' type multiple Genocchi zeta function as follows: where , and .
Define where , and .
From (4.3), we note that
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