Abstract and Applied AnalysisVolume 2012 (2012), Article ID 948050, 6 pagesdoi:10.1155/2012/948050

Research Article

On the Modified -Bernoulli Numbers of Higher Order with Weight

T. Kim,¹ J. Choi,² Y.-H. Kim,² and S.-H. Rim³

¹Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea²Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea³Department of Mathematics Education, Kyungpook National University, Daegu 702-701, Republic of Korea

Received 11 August 2011; Revised 24 November 2011; Accepted 13 December 2011

Academic Editor: Ferhan M. Atici

Copyright © 2012 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.

The purpose of this paper is to give some properties of the modified -Bernoulli numbers and polynomials of higher order with weight. In particular, by using the bosonic -adic -integral on , we derive new identities of -Bernoulli numbers and polynomials with weight.

1. Introduction

Let be a fixed odd 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 the algebraic closure of . Let be the set of natural numbers and . The -adic norm of is defined by . When one talks of a -extension, can be considered as an indeterminate, a complex number , or a -adic number . Throughout this paper we assume that and with so that .

Let be the space of uniformly differentiable functions on . For , the -adic -integral on is defined by Kim (see [1–3]) as follows: where is the -number of which is defined by .

From (1.1), we have where (see [2–4]).

As is well known, Bernoulli numbers are inductively defined by with the usual convention about replacing by (see [3, 5]).

In [2, 5, 6], the -Bernoulli numbers are defined by with the usual convention about replacing by . Note that . In the viewpoint of (1.4), we consider the modified -Bernoulli numbers with weight.

In this paper we study families of the modified -Bernoulli numbers and polynomials of higher order with weight. In particular, by using the multivariate -adic -integral on , we give new identities of the higher-order -Bernoulli numbers and polynomials with weight.

2. Modified -Bernoulli Numbers with Weight of Higher Order

For , let us consider the following modified -Bernoulli numbers with weight (see [1, 3]): From (2.1), we note that (see [1, 3]).

For and , by making use of the multivariate -adic -integral on , we consider the following modified -Bernoulli numbers with weight of order , : Note that and , where are the th ordinary Bernoulli numbers of order .

For , we have By (1.1), (2.3), and (2.4), we get Therefore, by (2.5), we obtain the following theorem.

Theorem 2.1. For , one has

Let us consider the modified -Bernoulli and polynomials with weight of order as follows: By the same method of (2.5), we obtain the following theorem.

Theorem 2.2. For , one has

By Theorem 2.2, we get Therefore, by (2.9), we obtain the following theorem.

Theorem 2.3. For , one has

From Theorem 2.3, we note that Thus, we have , where are the th Bernoulli numbers of order .

From (2.3) and (2.7), we can derive the following equations: Therefore, by (2.12), we obtain the following theorem.

Theorem 2.4. For and , one has

In particular,

From (1.2), we can derive the following integral: Continuing this process, we obtain By (2.16), we get Therefore, by (2.1) and (2.17), we obtain the following theorem.

Theorem 2.5. For and , one has

In an analogues manner as the previous investigation [7–10], we can define a further generalization of modified -Bernoulli numbers with weight. Let be the Dirichlet character with conductor . Then the generalized -Bernoulli numbers with weight attached to can be defined as follows: We expect to investigate these objects in future papers. This definition was also given in a previous paper (see [9]).

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.

References

S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionic -adic -integral representation on associated with weighted -Bernstein and -Genocchi polynomials,” Abstract and Applied Analysis, vol. 2011, Article ID 649248, 7 pages, 2011.

T. Kim, “-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.

T. Kim, “On the weighted -Bernoulli numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 207–215, 2011.

C. S. Ryoo and Y. H. Kim, “A numerical investigation on the structure of the roots of the twisted -Euler polynomials,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 131–141, 2009.

L. Carlitz, “-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, pp. 987–1000, 1948.

Y. Simsek, “Special functions related to Dedekind-type DC-sums and their applications,” Russian Journal of Mathematical Physics, vol. 17, no. 4, pp. 495–508, 2010.

T. Kim, “Power series and asymptotic series associated with the -analog of the two-variable -adic -function,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 186–196, 2005.

T. Kim, D. V. Dolgy, S. H. Lee, B. Lee, and S. H. Rim, “A note on the modified -Bernoulli numbers and polynomials with weight ,” Abstract and Applied Analysis, vol. 2011, Article ID 545314, 8 pages, 2011.

T. Kim, D. V. Dolgy, B. Lee, and S.-H. Rim, “Identities on the Weighted -Bernoulli numbers of higher order,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 918364, 6 pages, 2011.

H. M. Srivastava, T. Kim, and Y. Simsek, “-Bernoulli numbers and polynomials associated with multiple -zeta functions and basic -series,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 241–268, 2005.