Abstract and Applied Analysis

Abstract and Applied Analysis / 2008 / Article

Research Article | Open Access

Volume 2008 |Article ID 295307 | https://doi.org/10.1155/2008/295307

Lee-Chae Jang, "A New -Analogue of Bernoulli Polynomials Associated with -Adic -Integrals", Abstract and Applied Analysis, vol. 2008, Article ID 295307, 6 pages, 2008. https://doi.org/10.1155/2008/295307

A New -Analogue of Bernoulli Polynomials Associated with -Adic -Integrals

Academic Editor: Paul Eloe
Received14 May 2008
Accepted18 Jun 2008
Published12 Oct 2008

Abstract

We will study a new -analogue of Bernoulli polynomials associated with -adic -integrals. Furthermore, we examine the Hurwitz-type -zeta functions, replacing -adic rational integers with a -analogue for a -adic number with , which interpolate -analogue of Bernoulli polynomials.

1. Introduction

Let be a fixed odd prime number. Throughout this paper , , , and will, respectively, represent the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the -adic completion of the algebraic closure of . The -adic absolute value in is normalized so that and is a -adic number in with . We use the notation (cf. [113]) for all . Hence, . For a fixed odd positive integer with , let where lies in . For any , is known to be a distribution on (cf. [113]).

We say that is uniformly differentiable function at a point and denote this property by , if the difference quotients have a limit as (cf. [2, 6, 7]). The -adic -integral of a function was defined as

By using -adic -integrals on , it is well known that where . Then we note that the Bernoulli polynomials were defined as From (1.6) and (1.7), we have for all . We note that .

In Section 2, we study a -analogue of Bernoulli polynomials associated with -adic -integrals—simply, we say -Bernoulli polynomials. In Section 3, we examine Hurwitz-type   -zeta functions, replacing -adic rational integers with a -analogue for a -adic number with , which interpolate -analogue of Bernoulli polynomials.

2. A New -Analogue of Bernoulli Polynomials

In this section, from the view of (1.8), we can define a new -analogue of Bernoulli polynomials as follows: We note that are called the -Bernoulli numbers. Then we find some properties of -Bernoulli numbers and polynomials as follows.

Theorem 2.1. For , one has

Proof . From (1.5) with , we can find the following:

Theorem 2.2. For and being an odd positive integer with , one has

Proof . From (1.5), we can derive (2.4) as follows: since and for , and .
Let be the generating function of -Bernoulli polynomials as follows: From (2.2) and (2.7), we can obtain the following theorem.

Theorem 2.3. Let be as in the above generating function. Then, one has

Proof . By using (2.2) and (2.7), we can derive (2.8) as follows:

3. A New Formula for Hurwitz-Type   -Zeta Functions

In this section, we consider the generating functions which interpolate the -Bernoulli polynomials as follows: From (3.1), we directly obtain the following theorem.

Theorem 3.1. For each , one has

Proof . By the th differentiation on both sides of (3.1), we can derive (3.2) as follows:
We remark that for . From (3.2), we derive a -extension of Hurwitz-type zeta function as follows: for with and , we define Note that the functions are analytic on and they have simple pole at . From (3.2), (3.4), and (3.5), we can see that Hurwitz-type   -zeta functions interpolate -Bernoulli polynomials as follows.

Theorem 3.2. For each , one has

Acknowledgment

This paper was supported by Konkuk University in 2008.

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Copyright © 2008 Lee-Chae Jang. 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.


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