Abstract and Applied AnalysisVolume 2011 (2011), Article ID 545314, 8 pagesdoi:10.1155/2011/545314

Research Article

A Note on the Modified -Bernoulli Numbers and Polynomials with Weight

T. Kim,¹ D. V. Dolgy,² S. H. Lee,¹ B. Lee,³ and S. H. Rim⁴

¹Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea²Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea³Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, Republic of Korea⁴Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea

Received 13 July 2011; Accepted 2 October 2011

Academic Editor: Alberto d'Onofrio

Copyright © 2011 T. Kim et al. 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.

Abstract.

A systemic study of some families of the modified -Bernoulli numbers and polynomials with weight is presented by using the -adic -integration . The study of these numbers and polynomials yields an interesting -analogue related to 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, the complex number field, and the completion of the algebraic closure of . Let be the normalized exponential valuation of with . When one talks of -extension, is variously considered as an indeterminate, a complex , or a -adic number . If , one normally assumes . If , then we assume so that for .

The -number is defined by see [1–10].

We say that is a uniformly differentiable function at a point and denote this property by , if the difference quotients have a limit as . c.f. [11].

For , the -adic -integral on is defined by (see [3]).

From (1.3), we can easily derive the following: where , (see [5, 12]).

In [1, 2], Carlitz defined a set of numbers inductively by with the usual convention about replacing by .

These numbers are the -extension of ordinary Bernoulli numbers. But they do not remain finite when . So, Carlitz modified (1.5) as follows: with the usual convention of replacing by .

In [1], Carlitz also considered the extension of Carlitz’s -Bernoulli numbers as follows: with the usual convention of replacing by .

In this paper, we construct the modified -Bernoulli numbers with weight , which are different Carlitz’s -Bernoulli numbers, by using -adic -integral equation. From (1.4), we derive some interesting identities and relations on the modified -Bernoulli numbers and polynomials.

2. The Modified -Bernoulli Numbers and Polynomials with Weight

In this section, we assume . Now, we define the modified -Bernoulli numbers with weight ( ) as follows: Thus, by (2.1), we have Therefore, by (2.1) and (2.2), we obtain the following theorem.

Theorem 2.1. For , one has

Let us define the generating function of the modified -Bernoulli numbers with weight as follows: Then, by (2.3) and (2.4), we get In the viewpoint of (2.1), we define the modified -Bernoulli numbers with weight as follows: with the usual convention of replacing by .

From (2.6), we note that Therefore, by (2.7), we obtain the following theorem.

Theorem 2.2. For , one has

Let be the generating function of the modified -Bernoulli polynomials with weight .

Then, by (2.7), we get Therefore, by (2.9), we obtain the following corollary

Corollary 2.3. Let . Then one has

In particular, .

From Corollary 2.3, we can derive the following equation:

By (2.5) and (2.11), we get

Therefore, by (2.12), we obtain the following theorem.

Theorem 2.4. For , one has

By using (2.6), we obtain the following corollary.

Corollary 2.5. For , one has with the usual convention of replacing by .

From (1.4), we can derive the following equation:

Thus, by (1.6), (2.6), and (2.15), we get

Therefore, by (2.16), we obtain the following theorem.

Theorem 2.6. For , one has

From (2.6), we note that

Therefore, by (2.18), we obtain the following distribution relation for the modified -Bernoulli polynomials with weight .

Theorem 2.7. For , one has

To derive the relation of reflection symmetry of the modified -Bernoulli polynomials with weight , we evaluate the following -adic -integral on :

Therefore, by (2.20), we obtain the following reflection symmetry relation of the modified -Bernoulli polynomials with weight .

Theorem 2.8. For , one has

From (1.3), we note that and, by (2.6), we get

Let with . Then, by (2.12) and (2.23), we obtain the following theorem.

Theorem 2.9. For with , one has In particular,

Acknowledgments

The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2011.

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