Abstract and Applied Analysis

Abstract and Applied Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 848943 | 8 pages | https://doi.org/10.1155/2009/848943

Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

Academic Editor: Lance Littlejohn
Received04 Apr 2009
Accepted11 May 2009
Published06 Jul 2009

Abstract

By the properties of -adic invariant integral on , we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties of -adic invariant integral on , we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

1. Introduction

Let be a fixed prime number. Throughout this paper, the symbols , , , and will denote the ring of rational integers, the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. Let be the set of natural numbers and . Let be the normalized exponential valuation of with . Let be the space of uniformly differentiable function on . For , the -adic invariant integral on is defined as (see [1]). From the definition (1.1), we have Let , . Then we can derive the following equation from (1.2): (see [1]). It is well known that the ordinary Bernoulli polynomials are defined as (see [125]), and the Bernoulli number are defined as .

Let be a fixed positive integer. For , we set where lies in . It is easy to see that

In [14], the Witt's formula for the Bernoulli numbers are given by Let be the Dirichlet's character with conductor . Then the generalized Bernoulli polynomials attached to are defined as (see [22]), and the generalized Bernoulli numbers attached to , are defined as .

In this paper, we investigate the interesting identities of symmetry for the generalized Bernoulli numbers and polynomials attached to by using the properties of -adic invariant integral on . Finally, we will give relationship between the power sum polynomials and the generalized Bernoulli numbers attached to .

2. Symmetry of Power Sum and the Generalized Bernoulli Polynomials

Let be the Dirichlet character with conductor . From (1.3), we note that where are the th generalized Bernoulli numbers attached to . Now, we also see that the generalized Bernoulli polynomials attached to are given by By (2.1) and (2.2), we easily see that From (2.2), we have From (2.2), we can also derive Therefore, we obtain the following lemma.

Lemma 2.1. For , one has

We observe that Thus, we have Let us define the -adic functional as follows: By (2.8) and (2.9), we see that By using Taylor expansion in (2.10), we have That is, Let . Then we consider the following integral equation: From (2.7) and (2.10), we note that Let us consider the -adic functional as follows: Then we see that is symmetric in and , and By (2.15) and (2.16), we have

From the symmetric property of in and , we note that By comparing the coefficients on the both sides of (2.17) and (2.18), we obtain the following theorem.

Theorem 2.2. For , one has

Let in Theorem 2.2. Then we have By (2.14) and (2.16), we also see that From the symmetric property of in and , we can also derive the following equation: By comparing the coefficients on the both sides of (2.21) and (2.22), we obtain the following theorem.

Theorem 2.3. For , one has

Remark 2.4. Let in Theorem 2.3. Then we see that
If we take , then we have

Remark 2.5. Let be trivial character. Then we can easily derive the “multiplication theorem for Bernoulli polynomials’’ from Theorems 2.2 and 2.3 (see [14]).

Acknowledgment

The present research has been conducted by the research grant of the Kwangwoon University in 2009.

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Copyright © 2009 Taekyun 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.


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