International Journal of Mathematics and Mathematical Sciences

Volume 2011 (2011), Article ID 482840, 8 pages

http://dx.doi.org/10.1155/2011/482840

## Some Identities on the Twisted -Genocchi Numbers and Polynomials Associated with -Bernstein Polynomials

^{1}Department of Mathematics Education, Kyungpook National University, Daegu 702-701, Republic of Korea^{2}Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

Received 9 October 2011; Accepted 15 November 2011

Academic Editor: Taekyun Kim

Copyright © 2011 Seog-Hoon Rim and Sun-Jung Lee. 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 give some interesting identities on the twisted ()-Genocchi numbers and polynomials associated with -Bernstein polynomials.

#### 1. Introduction

Let be a fixed odd prime number. Throughout this paper, we always make use of the following notations: denotes the ring of rational integers, denotes the ring of -adic rational integer, denotes the ring of -adic rational numbers, and denotes the completion of algebraic closure of , respectively. Let be the set of natural numbers and . Let be the cyclic group of order and let The -adic absolute value is defined by , where ( and with () = () = () = 1). In this paper we assume that with as an indeterminate.

The -number is defined by (see [1–15]). Note that . Let be the space of uniformly differentiable function on . For , Kim defined the fermionic -adic -integral on as follows: (see [2–6, 8–15]). From (1.3), we note that (see [4–6, 8–12]), where for . For and , Kim defined the -Bernstein polynomials of the degree as follows: (see [13–15]). For and , let us consider the twisted ()-Genocchi polynomials as follows: Then, is called th twisted ()-Genocchi polynomials.

In the special case, and are called the th twisted ()-Genocchi numbers.

In this paper, we give the fermionic -adic integral representation of -Bernstein polynomial, which are defined by Kim [13], associated with twisted ()-Genocchi numbers and polynomials. And we construct some interesting properties of -Bernstein polynomials associated with twisted ()-Genocchi numbers and polynomials.

#### 2. On the Twisted -Genocchi Numbers and Polynomials

From (1.6), we note that We also have Therefore, we obtain the following theorem.

Theorem 2.1. *For and , one has
**
with usual convention about replacing by .**By (1.6) and (2.1) one gets
*

Therefore, we obtain the following theorem.

Theorem 2.2. *For and , one has
**From (1.5), one gets the following recurrence formula:
*

Therefore, we obtain the following theorem.

Theorem 2.3. *For and , one has
**
with usual convention about replacing by .*

From Theorem 2.3, we note that

Therefore, we obtain the following theorem.

Theorem 2.4. *For and , one has
*

*Remark 2.5. *We note that Theorem 2.4 also can be proved by using fermionic integral equation (1.4) in case of .

By (2.4) and Theorem 2.2, we get

Therefore, we obtain the following theorem.

Theorem 2.6. *For and , one has
*

Let . By Theorems 2.4 and 2.6, we get Therefore, we obtain the following corollary.

Corollary 2.7. *For and , one has
**By (1.5), we get the symmetry of -Bernstein polynomials as follows:
**
(see [11]).*

Thus, by Corollary 2.7 and (2.14), we get

From (2.15), we have the following theorem.

Theorem 2.8. *For and , one has
**For with , fermionic -adic invariant integral for multiplication of two -Bernstein polynomials on can be given by the following:
*

From Theorem 2.8 and (2.17), we have the following corollary.

Corollary 2.9. *For and , one has
**Let with . Then we get
*

From (2.19), we have the following theorem.

Theorem 2.10. *For and , one has
**Let with , fermionic -adic invariant integral for multiplication of two -Bernstein polynomials on can be given by the following:
*

From Theorem 2.10 and (2.21), we have the following corollary.

Corollary 2.11. *For and , one has
*

#### Acknowledgment

The authors would like to thank the anonymous referee for his/her excellent detail comments and suggestion.

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