Abstract and Applied Analysis

Volume 2011, Article ID 361484, 8 pages

http://dx.doi.org/10.1155/2011/361484

## Some Identities on the -Bernoulli Numbers and Polynomials with Weight 0

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

Received 16 August 2011; Accepted 27 September 2011

Academic Editor: Josef Diblík

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

Recently, Kim (2011) has introduced the -Bernoulli numbers with weight . In this paper, we consider the -Bernoulli numbers and polynomials with weight and give -adic -integral representation of Bernstein polynomials associated with -Bernoulli numbers and polynomials with weight . From these integral representation on , we derive some interesting identities on the -Bernoulli numbers and polynomials with weight .

#### 1. Introduction

Let be a fixed prime number. Throughout this paper, ,, and will denote the ring of -adic integers, the field of -adic rational numbers, and the completion of the algebraic closure of , respectively. Let be the set of natural numbers and .

Let be a -adic norm with , where and , . In this paper, we assume that with so that , and .

Let be the space of uniformly differentiable functions on . For , the -adic -integral on is defined by Kim as follows: (see [1–5]). For , let . From (1.1), we note that where , (see [3, 6, 7]). In the special case, , we get

Throughout this paper, we assume that .

The -Bernoulli numbers with weight are defined by Kim [8] as follows: with the usual convention about replacing with . From (1.4), we can derive the following equation: By (1.1), (1.4), and (1.5), we get The -Bernoulli polynomials with weight are defined by By (1.6) and (1.7), we easily see that

Let be the set of continuous functions on . For , the -adic analogue of Bernstein operator of order for is given by where (see [1, 9, 10]). For , the -adic Bernstein polynomials of degree are defined by for , (see [1, 10, 11]).

In this paper, we consider Bernstein polynomials to express the -adic -integral on and investigate some interesting identities of Bernstein polynomials associated with the -Bernoulli numbers and polynomials with weight 0 by using the expression of -adic -integral on of these polynomials.

#### 2. -Bernoulli Numbers with Weight 0 and Bernstein Polynomials

In the special case, , the -Bernoulli numbers with weight 0 will be denoted by . From (1.4), (1.5), and (1.6), we note that

It is easy to show that where are the th Frobenius-Euler numbers.

By (2.1) and (2.2), we get Therefore, we obtain the following theorem.

Theorem 2.1. *For , we have
**
where are the th Frobenius-Euler numbers.*

From (1.5), (1.6), and (1.7), we have with the usual convention about replacing with . By (1.7), the th -Bernoulli polynomials with weight 0 are given by From (2.6), we can derive the following function equation: Thus, by (2.7), we get that By the definition of -adic -integral on , we see that Therefore, by (2.8) and (2.9), we obtain the following theorem.

Theorem 2.2. *For , we have
**
In particular, , we get
*

From (2.5), we can derive the following equation: Therefore, by (2.12), we obtain the following theorem.

Theorem 2.3. *For with , we have
*

Taking the -adic -integral on for one Bernstein polynomials in (1.9), we get From the symmetry of Bernstein polynomials, we note that

Let . Then, by Theorem 2.3 and (2.15), we get

By comparing the coefficients on the both sides of (2.14) and (2.16), we obtain the following theorem.

Theorem 2.4. *For with , we have
**
In particular, when , we have
*

Let with . Then we see that For , we have

By comparing the coefficients on the both sides of (2.19) and (2.20), we obtain the following theorem.

Theorem 2.5. *For with , we have
**
In particular, when , we have
*

For , let with . By the same method above, we get From the binomial theorem, we note that

By comparing the coefficients on the both sides of (2.23) and (2.24), we obtain the following theorem.

Theorem 2.6. *For , let with . Then, we have
**
In particular, when , we have
*

#### Acknowledgment

The present research has been conducted by the Research Grant of Kwangwoon University in 2011.

#### References

- T. Kim, “$q$-Bernstein polynomials, $q$-Stirling numbers and $q$-Bernoulli polynomials,” http://arxiv.org/abs/1008.4547.
- T. Kim, “$q$-Volkenborn integration,”
*Russian Journal of Mathematical Physics*, vol. 9, no. 3, pp. 288–299, 2002. View at Google Scholar · View at Zentralblatt MATH - T. Kim, “$q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,”
*Russian Journal of Mathematical Physics*, vol. 15, no. 1, pp. 51–57, 2008. View at Google Scholar · View at Zentralblatt MATH - L.-C. Jang, “A study on the distribution of twisted $q$-Genocchi polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 18, no. 2, pp. 181–189, 2009. View at Google Scholar - 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. View at Publisher · View at Google Scholar - S.-H. Rim, J.-H. Jin, E.-J. Moon, and S.-J. Lee, “Some identities on the $q$-Genocchi polynomials of higher-order and $q$-Stirling numbers by the fermionic $p$-adic integral on ${\mathbb{Z}}_{p}$,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 860280, 14 pages, 2010. View at Publisher · View at Google Scholar - C. S. Ryoo, “Some identities of the twisted $q$-Euler numbers and polynomials associated with $q$-Bernstein polynomials,”
*Proceedings of the Jangjeon Matematical Society*, vol. 14, pp. 239–248, 2011. View at Google Scholar - T. Kim, “On the weighted $q$-Bernoulli numbers and polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 21, pp. 207–215, 2011. View at Google Scholar - S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionic $p$-adic $q$-integral representataion on ${\mathbb{Z}}_{p}$ assocatied with weighted $q$-Bernstein and $q$-Genocchi polynomials,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 649248, 7 pages, 2011. View at Publisher · View at Google Scholar - Y. Simsek, “Construction a new generating function of Bernstein type polynomials,”
*Applied Mathematics and Computation*. In press. - D. Erdal, J. J. Seo, and S. Araci, “New construction weighted $(h,q)$-Genocchi numbers and polynomials related to zeta type function,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 487490, 6 pages, 2011. View at Publisher · View at Google Scholar