Abstract and Applied Analysis

VolumeΒ 2011, Article IDΒ 123483, 9 pages

http://dx.doi.org/10.1155/2011/123483

## Some Identities of the Twisted -Genocchi Numbers and Polynomials with Weight and -Bernstein Polynomials with Weight

Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 7 July 2011; Accepted 22 August 2011

Academic Editor: JohnΒ Rassias

Copyright Β© 2011 H. Y. Lee 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 mathematicians have studied some interesting relations between -Genocchi numbers, -Euler numbers, polynomials, Bernstein polynomials, and -Bernstein polynomials. In this paper, we give some interesting identities of the twisted -Genocchi numbers, polynomials, and -Bernstein polynomials with weighted .

#### 1. Introduction

Throughout this paper, let be a fixed odd prime number. The symbols , , and denote the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of . Let be the set of natural numbers and let . As a well-known definition, the -adic absolute value is given by , where with . When one talks of -extension, is variously considered as an indeterminate, a complex number , or a -adic number . In this paper we assume that with .

We assume that is the space of the uniformly differentiable function on . For , Kim defined the fermionic -adic -integral on as follows: For , let be translation. As a well known equation, by (1.1), we have compared [1β4]. Throughout this paper we use the notation: (cf. [1β16]). for any with in the present -adic case. To investigate relation of the twisted -Genocchi numbers and polynomials with weight and the Bernstein polynomials with weight , we will use useful property for as follows;

The twisted -Genocchi numbers and polynomials with weight are defined by the generating function as follows, respectively: In the special case, , are called the th twisted -Genocchi numbers with weight (see [9]).

Let be the cyclic group of order and let see [9, 12β15].

Kim defined the -Bernstein polynomials with weight of degree as follows: compare [4, 7].

In this paper, we investigate some properties for the twisted -Genocchi numbers and polynomials with weight . By using these properties, we give some interesting identities on the twisted -Genocchi polynomials with weight and -Bernstein polynomials with weight .

#### 2. Some Identities on the Twisted -Genocchi Polynomials with Weight and -Bernstein Polynomials with Weight

From (1.8), we can derive the following recurrence formula for the twisted -Genocchi numbers with weight : with usual convention about replacing by .

By (1.5), we easily get By (2.4), we obtain the theorem below.

Theorem 2.1. *Let . For , one has
*

*By (2.1), (2.2), and (2.3) we note that *

*Therefore, by (2.6), we obtain the theorem below.*

*Theorem 2.2. For with , one has
*

*By (1.6) and Theorem 2.2,
Hence, we obtain the corollary below.*

*Corollary 2.3. For , one has
*

*By fermionic integral on ,Theorems 2.1 and 2.2, we note that*

*Therefore, we have the theorem below.*

*Theorem 2.4. For with , one has
*

*By (1.4), Theorem 2.4, we take the fermionic -adic invariant integral on for one q-Bernstein polynomials as follows:
*

*By symmetry of -Bernstein polynomials with weight of degree , we get the following formula;
*

*Therefore, by (2.12) and (2.13), we have the theorem below.*

*Theorem 2.5. For with , one has
Also, we note that
*

*Therefore, we have the theorem below.*

*Theorem 2.6. For with , one has
*

*By (2.11) and Theorem 2.6, we have the theorem below.*

*Theorem 2.7. Let with . Then one has
*

*Let with . Then we get *

*Therefore, we obtain the theorem below.*

*Theorem 2.8. For , one has
*

*By simple calculation, we easily see that
Therefore, by (2.20) and Theorem 2.8, we obtain the theorem below.*

*Theorem 2.9. Let with . Then one has
*

*For , , and let , then by the symmetry of -Bernstein polynomials with weight , we see that *

*Therefore, we have the theorem below.*

*Theorem 2.10. For with , one has
where .*

*In the same manner as in (2.15), we can get the following relation:
where with .*

*By Theorem 2.10 and (2.13), we have the following corollary.*

*Corollary 2.11. Let . For with , one has
where .*

*References*

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