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

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 640152 (2009)

## 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 .

## 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 [1–24]). 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

(see [11–19]). From (1.2), we note that

where and . Let . Then we can derive the following equation from (1.3):

(see [1–11]). 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 [1–20, 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

By (2.1) and (2.2), we have

(see [1–19, 25]). For , we obtain that

where . Thus, we have

Then

Let us define the -adic function as follows:

(see [25]). By (2.7) and (2.8), we see that

(see [25]). Thus, we have

This means that

(see [25]).

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

where

In (2.13), we note that is symmetric in . From (2.13), we derive

It is easy to see that

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 [25]).

We also calculate that

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 [25]).

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## Acknowledgment

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

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Kim, T., Jang, LC., Kim, YH. *et al.* On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order.
*J Inequal Appl* **2009**, 640152 (2009). https://doi.org/10.1155/2009/640152

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