`Abstract and Applied AnalysisVolume 2008, Article ID 304539, 13 pageshttp://dx.doi.org/10.1155/2008/304539`
Research Article

Multivariate -Adic Fermionic -Integral on and Related Multiple Zeta-Type Functions

1Department of Mathematics, Kyungnam University, Masan 631-701, South Korea
2Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea

Received 14 April 2008; Accepted 27 May 2008

Copyright © 2008 Min-Soo Kim 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

In 2008, Jang et al. constructed generating functions of the multiple twisted Carlitz's type -Bernoulli polynomials and obtained the distribution relation for them. They also raised the following problem: “are there analytic multiple twisted Carlitz's type -zeta functions which interpolate multiple twisted Carlitz's type -Euler (Bernoulli) polynomials?” The aim of this paper is to give a partial answer to this problem. Furthermore we derive some interesting identities related to twisted -extension of Euler polynomials and multiple twisted Carlitz's type -Euler polynomials.

1. Introduction, Definitions, and Notations

Let be an odd prime. and will always denote, respectively, the ring of -adic integers, the field of -adic numbers, and the completion of the algebraic closure of Let ( is the field of rational numbers) denote the -adic valuation of normalized so that The absolute value on will be denoted as and for We let . A -adic integer in is sometimes called a -adic unit. For each integer , will denote the multiplicative group of the primitive th roots of unity in SetThe dual of in the sense of -adic Pontrjagin duality, is the direct limit (under inclusion) of cyclic groups of order with the discrete topology.

When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number If then we normally assume so that for If then we assume that

Letwhere lies in .

We use the following notation:Hencefor any with in the present -adic case. The distribution is given as(cf. [19]). For the ordinary -adic distribution defined bywe seeWe say that is a uniformly differentiable function at a point we write if the difference quotienthas a limit as . Also we use the following notation: (cf.[15]).

In [13], Kim gave a detailed proof of fermionic -adic -measures on He treated some interesting formulae-related -extension of Euler numbers and polynomials; and he defined fermionic -adic -measures on as follows:By using the fermionic -adic -measures, he defined the fermionic -adic -integral on as follows:for (cf. [13]). Observe thatFrom (1.12), we obtainwhere By substituting into (1.13), classical Euler numbers are defined by means of the following generating function:These numbers are interpolated by the Euler zeta function which is defined as follows:(cf. [19]). From (1.12), we also obtainwhere By substituting into (1.13), -Euler numbers are defined by means of the following generating function:These numbers are interpolated by the Euler -zeta function which is defined as follows:(cf. [4]).

In [6], Ozden and Simsek defined generating function of -Euler numbers bywhich are different from (1.17). But we observe that all these generating functions were obtained by the same fermionic -adic -measures on and the fermionic -adic -integrals on

In this paper, we define a multiple twisted Carlitz's type -zeta functions, which interpolated multiple twisted Carlitz's type -Euler polynomials at negative integers. This result gave us a partial answer of the problem proposed by Jang et al. [10], which is given by: “Are there analytic multiple twisted Carlitz's type -zeta functions which interpolate multiple twisted Carlitz's type -Euler (Bernoulli) polynomials?

2. Preliminaries

In [10], Jang and Ryoo defined -extension of Euler numbers and polynomials of higher order and studied multivariate -Euler zeta functions. They also derived sums of products of -Euler numbers and polynomials by using ferminonic -adic -integral.

In [5, 7], Ozden et al. defined multivariate Barnes-type Hurwitz -Euler zeta functions and -functions. They also gave relation between multivariate Barnes-type Hurwitz -Euler zeta functions and multivariate -Euler -functions.

In this section, we consider twisted -extension of Euler numbers and polynomials of higher order and study multivariate twisted Barnes-type Hurwitz -Euler zeta functions and -functions.

Let denote the space of all uniformly (or strictly) differentiable -valued functions on For the -adic -integral on is defined by(cf. [3]). If thenFor a fixed positive integer with we setFor (cf. [2]).

We set in (2.2) and (2.4). Then we havewhere are the twisted Euler polynomials of order From (2.5), we note that

We give an application of the multivariate -deformed -adic integral on in the fermionic sense related to [3]. LetBy substitutinginto (2.1), we define twisted -extension of Euler numbers of higher order by means of the following generating function:Then we haveFrom (2.9), we obtainwhere is called twisted -extension of Euler polynomials of higher order (cf. [11]). We note that if then and (cf. [6]). We also note that

The twisted -extension of Euler polynomials of higher order, is defined by means of the following generating function:where From these generating functions of twisted -extension of Euler polynomials of higher order, we construct twisted multiple -Euler zeta functions as follows.

For and with we defineBy the th differentiation on both sides of (2.13) at we obtain the followingfor

From (2.14) and (2.15), we arrive at the following

We setLet be Dirichlet's character with odd conductor We define twisted -extension of generalized Euler polynomials of higher order by means of the following generating function (cf. [11]):Note thatsince This allows us to rewrite (2.18) asBy applying the th derivative operator in the above equation, we havefor

From these generating functions of twisted -extension of generalized Euler polynomials of higher order, we construct twisted multiple -Euler -functions as follows. For and with we defineFrom (2.22) and (2.23), we arrive at the following

Let and with is an odd integer and where Then twisted partial multiple -Euler -functions are as follows:For substituting with is odd into (2.25), we haveThen we obtainBy using (2.12) and (2.27) and by substituting we getTherefore, we modify twisted partial multiple -Euler zeta functions as follows:Let be a Dirichlet character with conductors and From (2.23) and (2.27), we have

3. The Multiple Twisted Carlitz's Type -Euler Polynomials and -Zeta Functions

Let us consider the multiple twisted Carlitz's type -Euler polynomials as follows:(cf. [1, 3]). These can be written asWe may now mention the following formulae which are easy to prove:From (3.2), we can derive generating function for the multiple twisted Carlitz's type -Euler polynomials as follows:Also, an obvious generating function for the multiple twisted Carlitz's type -Euler polynomials is obtained, from (3.2), by

From now on, we assume that with From (3.2) and (3.4), we note thatThus we can define the multiple twisted Carlitz's type -zeta functions as follows:

In [12, Proposition 3], Yamasaki showed that the series converges absolutely for Re and it can be analytically continued to the whole complex plane Note that if then

In [13], Wakayama and Yamasaki studied -analogue of the Hurwitz zeta functiondefined by the -series with two complex variable : and special values at nonpositive integers of the -analogue of the Hurwitz zeta function.

Therefore, by the th differentiation on both sides of (3.6) at we obtain the following:for

From (3.7), (3.8), and (3.12), we have (3.13) which shows that the multiple twisted Carlitz's type -zeta functions interpolate the multiple twisted Carlitz's type -Euler numbers and polynomials. For we havewhere and

Thus, we derive the analytic multiple twisted Carlitz's type -zeta functions which interpolate multiple twisted Carlitz's type -Euler polynomials. This gives a part of the answer to the question proposed in [10].

4. Remarks

For nonnegative integers and we define the -binomial coefficient bywhere for and For it holds that(cf. [12, Lamma 2.3]). From (3.8), it is easy to see thatWe set for The following identity has been studied in [12]:where is a function of defined byfor and By using (3.9), (4.3), and (4.5), we haveand soThe values of at are given explicitly as follows:

Acknowledgment

This work is supported by Kyungnam University Foundation Grant no. 2007.

References

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