Abstract and Applied AnalysisVolume 2011 (2011), Article ID 392025, 14 pagesdoi:10.1155/2011/392025

Research Article

On the -Bernoulli Numbers and Polynomials with Weight

T. Kim and J. Choi

Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 13 May 2011; Accepted 26 July 2011

Academic Editor: Elena Litsyn

Copyright © 2011 T. Kim and J. Choi. 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.

We present a systemic study of some families of higher-order -Bernoulli numbers and polynomials with weight . From these studies, we derive some interesting identities on the -Bernoulli numbers and polynomials with weight .

1. Introduction

Let be a fixed odd prime number. Throughout this paper, , , and will denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. The -adic norm of is defined as , where with , and . Let and be the set of natural numbers and integers, respectively, . Let with . The notation of -number is defined by and , (see [1–13]).

As the well known definition, the Bernoulli polynomials are defined by In the special case, , are called the th Bernoulli numbers. That is, the recurrence formula for the Bernoulli numbers is given by with the usual convention about replacing with .

In [1, 2], -extension of Bernoulli numbers are defined by Carlitz as follows: with the usual convention about replacing with .

By (1.2) and (1.3), we get . In this paper, we assume that .

In [7], the -Bernoulli numbers with weight are defined by Kim as follows: with the usual convention about replacing with .

Let be the space of uniformly differentiable functions on . For , the -adic -integral on is defined as (see[4, 5]). From (1.5), we note that where and .

By (1.4), (1.5), and (1.6), we set (see[7]). The -Bernoulli polynomials are also given by

The purpose of this paper is to derive a new concept of higher-order -Bernoulli numbers and polynomials with weight from the fermionic -adic -integral on . Finally, we present a systemic study of some families of higher-order -Bernoulli numbers and polynomials with weight .

2. Higher Order -Bernoulli Numbers with Weight

Let and in this paper. For and , we consider the expansion of higher-order -Bernoulli polynomials with weight as follows:

From (2.1), we note that where and .

Therefore, we obtain the following theorem.

Theorem 2.1. For and , we have

In the special case, , are called the th higher order -Bernoulli numbers with weight .

From (2.1) and (2.2), we can derive By Theorem 2.1 and (2.4), we get From (2.1), we have

Thus, we obtain the following theorem.

Theorem 2.2. For , we have

It is easy to show that

3. Polynomials

In this section, we consider the polynomials as follows:

From (3.1), we can easily derive the following equation: By (3.1) and (3.2), we get Therefore, by (3.3) and (3.4), we obtain the following theorem.

Theorem 3.1. For , we have Moreover,

Let . Then, we have Thus, by (3.1) and (3.7), we obtain the following theorem.

Theorem 3.2. For , and , we have

From (3.1), we note that

4. Polynomials

For , let us define weighted -Bernoulli polynomials as follows: By (4.1), we easily see that Therefore, by (4.2), we obtain the following theorem.

Theorem 4.1. For and , we have

From (4.1), we can derive the following equation: By (4.4), we easily get From (4.1), we have where .

By (4.6), we get the following recurrence formula: with the usual convention about replacing with .

From (1.6), we note that For , by (4.8), we have If , then we get Let . By (4.9), we get From (4.6) and (4.11), we note that If we take in (4.12), then we have Therefore, by (4.12) and (4.13), we obtain the following theorem.

Theorem 4.2. For , we have

By (4.7) and Theorem 4.2, we obtain the following corollary.

Corollary 4.3. For , we have with the usual convention about replacing with .

From (4.1), we have It is not difficult to show that Therefore, by (4.17), we obtain the following theorem.

Theorem 4.4. For , we have

For in Theorem 4.4, we get Therefore, by (4.19), we obtain the following corollary.

Corollary 4.5. For and with , we have

Let . By (4.1), we see that By (4.1) and (4.21), we obtain the following equation: where and .

5. Polynomials and

From (2.1), we note that From (5.1), we have Therefore, by (5.3), we obtain the following theorem.

Theorem 5.1. For and , we have

It is easy to show that Thus, by (5.5), we obtain the following proposition.

Proposition 5.2. For , we have

For , we get Thus, we obtain Equation (5.8) is multiplication formula for the -Bernoulli polynomials of order with weight .

Let us define . Then we see that Therefore, by (5.9) and (5.10), we obtain the following theorem.

Theorem 5.3. For and , we have

Let in Theorem 5.3. Then we see that From Theorem 5.1, we can derive the following equation: By (5.9), we easily get From the definition of -adic -integral on , we note that Thus, by (5.15), we get By the definition of polynomial , we see that with the usual convention about replacing with .

Let in (5.13). Then we have

Acknowledgments

The present research has been conducted by the Research Grant of Kwangwoon University in 2011. The authors would like to express their gratitude to referees for their valuable suggestions and comments.

References

L. Carlitz, “Expansions of -Bernoulli numbers,” Duke Mathematical Journal, vol. 25, pp. 355–364, 1958.

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

M. Can, M. Cenkci, V. Kurt, and Y. Simsek, “Twisted Dedekind type sums associated with Barnes' type multiple Frobenius-Euler l-functions,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 135–160, 2009.

T. Kim, “On a -analogue of the -adic log gamma functions and related integrals,” Journal of Number Theory, vol. 76, no. 2, pp. 320–329, 1999.

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

T. Kim, “-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,” Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008.

T. Kim, “On the weighted -Bernoulli numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, pp. 207–215, 2011.

T. Kim, J. Choi, and Y. H. Kim, “-Bernstein polynomials associated with -Stirling numbers and Carlitz's -Bernoulli numbers,” Abstract and Applied Analysis, Article ID 150975, 11 pages, 2010.

T. Kim, “Some identities on the -Euler polynomials of higher order and -Stirling numbers by the fermionic -adic integral on ,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009.

H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on -Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.

Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–278, 2008.

S.-H. Rim, E. J. Moon, S. J. Lee, and J. H. Jin, “On the -Genocchi numbers and polynomials associated with -zeta function,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 3, pp. 261–267, 2009.

H. Ozden, Y. Simsek, S. H. Rim, and I. N. Cangul, “A note on -adic -Euler measure,” Advanced Studies in Contemporary Mathematics, vol. 14, pp. 233–239, 2007.