• Research Article
• Open Access

# On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order

Journal of Inequalities and Applications20092009:640152

https://doi.org/10.1155/2009/640152

• Received: 5 June 2009
• Accepted: 5 August 2009
• Published:

## Abstract

We give some interesting relationships between the power sums and the generalized Bernoulli numbers attached to of higher order using multivariate -adic invariant integral on .

## Keywords

• Positive Integer
• Natural Number
• Prime Number
• Rational Number
• Differentiable Function

## 1. Introduction

Let be a fixed prime number. Throughout this paper, the symbols , , , and 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 (see ). Let be the space of uniformly differentiable function on . Let be a fixed positive integer. For , let
where lies in . For , the -adic invariant integral on is defined as
where and . Let . Then we can derive the following equation from (1.3):
(see ). Let be the Dirichlet's character with conductor . Then the generalized Bernoulli polynomials attached to are defined as

and the generalized Bernoulli numbers attached to , , are defined as (see [120, 25]). The purpose of this paper is to derive some identities of symmetry for the generalized Bernoulli polynomials attached to of higher order.

## 2. Symmetric Properties for the Generalized Bernoulli Polynomials of Higher Order

Let be the Dirichlet's character with conductor . Then we note that
where are the th generalized Bernoulli numbers attached to (see [7, 9, 15, 25]). Now we also see that the generalized Bernoulli polynomials attached to are given by
(see [119, 25]). For , we obtain that
where . Thus, we have
Let us define the -adic function as follows:

(see ).

The generalized Bernoulli polynomials attached to of order , which is denoted by , are defined as
Then the values of at are called the generalized Bernoulli numbers attached to of order . When , the polynomials of numbers are called the generalized Bernoulli polynomials or numbers attached to . Let . Then we set
In (2.13), we note that is symmetric in . From (2.13), we derive

From (2.16), we note that

By the symmetry of in and , we see that

By comparing the coefficients on the both sides of (2.17) and (2.18), we see the following theorem.

Theorem 2.1.

For , one has

Remark 2.2.

Let and in (1.4). Then we have

(see ).

From the symmetric property of in and , we derive

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 and in (2.23). We have

(see ).

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

(1)
Division of General Education-Mathematics, Kwangwoon University, Seoul, 139-701, South Korea
(2)
Department of Mathematics and Computer Science, Konkook University, Chungju, 139-701, South Korea
(3)
Department of General Education, Kookmin University, Seoul, 136-702, South Korea

## References 