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# Multivariate Twisted -Adic -Integral on Associated with Twisted -Bernoulli Polynomials and Numbers

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 579509 (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.

## 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 .

For let be the -adic locally constant space defined by

where is the cyclic group of order

Let be the space of uniformly differentiable function on

For the -adic invariant -integral on is defined as

It is well known that the twisted -Bernoulli polynomials of order are defined as

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 this section, we assume that with For , we define the -numbers as

Note that .

Let us define

where Note that

Now we construct the twisted -adic -integral on as follows:

where From the definition of the twisted -adic -integral on , we can consider the twisted -Bernoulli polynomials and numbers of order as follows:

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

If we take and in (2.4), we can easily see that

compare with [4].

Theorem 2.1.

For and we have

Moreover, if we take in Theorem 2.1, then we have the following identity for the twisted -Bernoull numbers

From the definition of multivariate twisted -adic -integral, we also see that

Corollary 2.2.

For and , one obtains

Note that

We have

It is easy to see that

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

Theorem 2.3.

For and one has

Now we consider the modified extension of the twisted -Bernoulli polynomials of order as follows:

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

From (2.14), we derive that

Therefore, we obtain the following theorem.

Theorem 2.4.

For and one has

Now, we define as follows:

By (2.17), we can see that

Therefore, we obtain the following theorem.

Theorem 2.5.

For and one has

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

## References

Jang L-C: Multiple twisted -Euler numbers and polynomials associated with -adic -integrals.

*Advances in Difference Equations*2008, -11.Kim T: -Volkenborn integration.

*Russian Journal of Mathematical Physics*2002, 9(3):288–299.Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.

*Russian Journal of Mathematical Physics*2008, 15(1):51–57.Kim T: Sums of products of -Bernoulli numbers.

*Archiv der Mathematik*2001, 76(3):190–195. 10.1007/s000130050559

## 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.

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**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### Cite this article

Rim, SH., Moon, EJ., Lee, SJ. *et al.* Multivariate Twisted -Adic -Integral on Associated with Twisted -Bernoulli Polynomials and Numbers.
*J Inequal Appl* **2010, **579509 (2010). https://doi.org/10.1155/2010/579509

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

### Keywords

- Bernoulli Polynomial
- Constant Space