# A Note on Symmetric Properties of the Twisted -Bernoulli Polynomials and the Twisted Generalized -Bernoulli Polynomials

- L.-C. Jang
^{1}, - H. Yi
^{2}Email author, - K. Shivashankara
^{3}, - T. Kim
^{4}, - Y. H. Kim
^{4}and - B. Lee
^{5}

**2010**:801580

https://doi.org/10.1155/2010/801580

© L.-C. Jang et al. 2010

**Received: **11 September 2009

**Accepted: **31 May 2010

**Published: **24 June 2010

## Abstract

We define the twisted -Bernoulli polynomials and the twisted generalized -Bernoulli polynomials attached to of higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain some relationships between twisted -Bernoulli numbers and polynomials and between twisted generalized -Bernoulli numbers and polynomials.

## 1. Introduction

Let be a fixed prime number. Throughout this paper , and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of . Let be the normalized exponential valuation of with When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number . If one normally assumes If then we assume so that for (cf. [1–32]).

In this paper, we define the twisted -Bernoulli polynomials and the twisted generalized -Bernoulli polynomials attached to of higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain some relationships between the twisted -Bernoulli numbers and polynomials and between the twisted generalized -Bernoulli numbers and polynomials attached to of higher order.

## 2. The Twisted -Bernoulli Polynomials

By comparing the coefficients of on both sides of (2.16) and (2.17), we obtain the following theorem.

Theorem 2.1.

where is the binomial coefficient.

From Theorem 2.1, if we take , then we have the following corollary.

Corollary 2.2.

where is the binomial coefficient.

By comparing the coefficients of on both sides of (2.20) and (2.21), we obtain the following theorem.

Theorem 2.3.

We note that by setting in Theorem 2.3, we get the following multiplication theorem for the twisted -Bernoulli polynomials.

Theorem 2.4.

Remark 2.5.

[18], Kim suggested open questions related to finding symmetric properties for Carlitz -Bernoulli numbers. In this paper, we give the symmetric property for -Bernoulli numbers in the viewpoint to give the answer of Kim's open questions.

## 3. The Twisted Generalized Bernoulli Polynomials Attached to of Higher Order

By comparing the coefficients on both sides of (3.20) and (3.21), we see the following theorem.

Theorem 3.1.

Remark 3.2.

By comparing the coefficients on both sides of (3.24) and (3.26), we obtain the following theorem.

Theorem 3.3.

Remark 3.4.

Remark 3.5.

In our results for , we can also derive similar results, which were treated in [27]. In this paper, we used the -adic integrals to derive the symmetric properties of the -Bernoulli polynomials. By using the symmetric properties of -adic integral on , we can easily derive many interesting symmetric properties related to Bernoulli numbers and polynomials.

## Declarations

### Acknowledgments

The authors express Their sincere gratitude to referees for their valuable suggestions and comments. This work has been conducted by the Research Grant of Kwangwoon University in 2010.

## Authors’ Affiliations

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