Research Article | Open Access

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

#### 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. [1–13]) for all . Hence, . For a fixed odd positive integer with , letwhere lies in . For any ,is known to be a distribution on (cf. [1–13]).

We say that is uniformly differentiable function at a point and denote this property by , if the difference quotientshave 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 thatwhere . Then we note that the Bernoulli polynomials were defined asFrom (1.6) and (1.7), we havefor 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 andfor , 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 thatfor .
From (3.2), we derive a -extension of
Hurwitz-type zeta function as follows: for with and ,
we defineNote 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

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.