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

  • Seog-Hoon Rim1Email author,
  • Eun-Jung Moon1,
  • Sun-Jung Lee1 and
  • Jeong-Hee Jin1
Journal of Inequalities and Applications20102010:579509

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

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

Bernoulli PolynomialConstant Space

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
(1.1)

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
(1.2)

compare with [13].

It is well known that the twisted -Bernoulli polynomials of order are defined as
(1.3)

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
(2.1)

Note that .

Let us define
(2.2)

where Note that

Now we construct the twisted -adic -integral on as follows:
(2.3)
where From the definition of the twisted -adic -integral on , we can consider the twisted -Bernoulli polynomials and numbers of order as follows:
(2.4)

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
(2.5)

compare with [4].

Theorem 2.1.

For and we have
(2.6)
Moreover, if we take in Theorem 2.1, then we have the following identity for the twisted -Bernoull numbers
(2.7)
From the definition of multivariate twisted -adic -integral, we also see that
(2.8)

Corollary 2.2.

For and , one obtains
(2.9)
Note that
(2.10)
We have
(2.11)
It is easy to see that
(2.12)

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

Theorem 2.3.

For and one has
(2.13)
Now we consider the modified extension of the twisted -Bernoulli polynomials of order as follows:
(2.14)

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
(2.15)

Therefore, we obtain the following theorem.

Theorem 2.4.

For and one has
(2.16)
Now, we define as follows:
(2.17)
By (2.17), we can see that
(2.18)

Therefore, we obtain the following theorem.

Theorem 2.5.

For and one has
(2.19)

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

(1)
Department of Mathematics, Kyungpook National University, Daegu, Republic of Korea

References

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

Copyright

© Seog-Hoon Rim et al. 2010

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.

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