Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 392025, 14 pages

http://dx.doi.org/10.1155/2011/392025

## On the -Bernoulli Numbers and Polynomials with Weight

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

Received 13 May 2011; Accepted 26 July 2011

Academic Editor: ElenaΒ Litsyn

Copyright Β© 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 [1β13]).

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, βExpansions of $q$-Bernoulli numbers,β
*Duke Mathematical Journal*, vol. 25, pp. 355β364, 1958. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - L. Carlitz, β$q$-Bernoulli numbers and polynomials,β
*Duke Mathematical Journal*, vol. 15, pp. 987β1000, 1948. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - M. Can, M. Cenkci, V. Kurt, and Y. Simsek, βTwisted Dedekind type sums associated with Barnes' type multiple Frobenius-Euler
*l*-functions,β*Advanced Studies in Contemporary Mathematics*, vol. 18, no. 2, pp. 135β160, 2009. View at Google Scholar - T. Kim, βOn a $q$-analogue of the $p$-adic log gamma functions and related integrals,β
*Journal of Number Theory*, vol. 76, no. 2, pp. 320β329, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - T. Kim, β$q$-volkenborn integration,β
*Russian Journal of Mathematical Physics*, vol. 9, no. 3, pp. 288β299, 2002. View at Google Scholar Β· View at Zentralblatt MATH - T. Kim, β$q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,β
*Russian Journal of Mathematical Physics*, vol. 15, no. 1, pp. 51β57, 2008. View at Google Scholar Β· View at Zentralblatt MATH - T. Kim, βOn the weighted $q$-Bernoulli numbers and polynomials,β
*Advanced Studies in Contemporary Mathematics*, vol. 21, pp. 207β215, 2011. View at Google Scholar - T. Kim, J. Choi, and Y. H. Kim, β$q$-Bernstein polynomials associated with $q$-Stirling numbers and Carlitz's $q$-Bernoulli numbers,β
*Abstract and Applied Analysis*, Article ID 150975, 11 pages, 2010. View at Google Scholar - T. Kim, βSome identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on ${\mathbb{Z}}_{p}$,β
*Russian Journal of Mathematical Physics*, vol. 16, no. 4, pp. 484β491, 2009. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - H. Ozden, I. N. Cangul, and Y. Simsek, βRemarks on $q$-Bernoulli numbers associated with Daehee numbers,β
*Advanced Studies in Contemporary Mathematics*, vol. 18, no. 1, pp. 41β48, 2009. View at Google Scholar - Y. Simsek, βGenerating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,β
*Advanced Studies in Contemporary Mathematics*, vol. 16, no. 2, pp. 251β278, 2008. View at Google Scholar Β· View at Zentralblatt MATH - S.-H. Rim, E. J. Moon, S. J. Lee, and J. H. Jin, βOn the $q$-Genocchi numbers and polynomials associated with $q$-zeta function,β
*Proceedings of the Jangjeon Mathematical Society*, vol. 12, no. 3, pp. 261β267, 2009. View at Google Scholar - H. Ozden, Y. Simsek, S. H. Rim, and I. N. Cangul, βA note on $p$-adic $q$-Euler measure,β
*Advanced Studies in Contemporary Mathematics*, vol. 14, pp. 233β239, 2007. View at Google Scholar