- Seog-Hoon Rim
^{1}Email author, - Eun-Jung Moon
^{1}, - Sun-Jung Lee
^{1}and - Jeong-Hee Jin
^{1}

**2010**:579509

https://doi.org/10.1155/2010/579509

© Seog-Hoon Rim et al. 2010

**Received: **19 June 2010

**Accepted: **2 October 2010

**Published: **11 October 2010

## Abstract

Recently, many authors have studied twisted -Bernoulli polynomials by using the -adic invariant -integral on . In this paper, we define the twisted -adic -integral on and extend our result to the twisted -Bernoulli polynomials and numbers. Finally, we derive some various identities related to the twisted -Bernoulli polynomials.

## Keywords

## 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, the complex number field, and the completion of the algebraic closure of respectively. Let be the set of natural numbers and Let be the normalized exponential valuation of with

When one talks of -extension, is variously considered as an indeterminate, a complex or -adic number If one normally assumes that If then we assume that .

where is the cyclic group of order

Let be the space of uniformly differentiable function on

and are called the twisted -Bernoulli numbers of order When the polynomials and numbers are called the twisted -Bernoulli polynomials and numbers, respectively. When and the polynomials and numbers are called the twisted Bernoulli polynomials and numbers, respectively. When and the polynomials and numbers are called the ordinary Bernoulli polynomials and numbers, respectively.

Many authors have studied the twisted -Bernoulli polynomials by using the properties of the -adic invariant -integral on (cf. [4]). In this paper, we define the twisted -adic -integral on and extend our result to the twisted -Bernoulli polynomials and numbers. Finally, we derive some various identities related to the twisted -Bernoulli polynomials.

## 2. Multivariate Twisted -Adic -Integral on Associated withTwisted -Bernoulli Polynomials

In the special case in (2.4), are called the twisted -Bernoulli numbers of order .

compare with [4].

Theorem 2.1.

Corollary 2.2.

By (2.11) and (2.12), we obtain the following theorem.

Theorem 2.3.

In the special case we write which are called the modified extension of the twisted -Bernoulli numbers of order

Therefore, we obtain the following theorem.

Theorem 2.4.

Therefore, we obtain the following theorem.

Theorem 2.5.

In (2.19), we can see the relations between the binomial coefficients and the modified extension of the twisted -Bernoulli polynomials of order

## Declarations

### Acknowledgments

The authors would like to thank the anonymous referee for his/her excellent detail comments and suggestions. This Research was supported by Kyungpook National University Research Fund, 2010.

## Authors’ Affiliations

## References

- Jang L-C: Multiple twisted -Euler numbers and polynomials associated with -adic -integrals.
*Advances in Difference Equations*2008, -11.Google Scholar - Kim T: -Volkenborn integration.
*Russian Journal of Mathematical Physics*2002, 9(3):288–299.MathSciNetMATHGoogle Scholar - Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.
*Russian Journal of Mathematical Physics*2008, 15(1):51–57.MathSciNetView ArticleMATHGoogle Scholar - Kim T: Sums of products of -Bernoulli numbers.
*Archiv der Mathematik*2001, 76(3):190–195. 10.1007/s000130050559MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.