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Journal of Applied Mathematics
Volume 2011, Article ID 609054, 10 pages
http://dx.doi.org/10.1155/2011/609054
Research Article

A Note on Some Properties of the Weighted -Genocchi Numbers and Polynomials

Department of Mathematics and Computer Science, Konkuk University, Chungju 280-701, Republic of Korea

Received 30 July 2011; Revised 23 September 2011; Accepted 23 September 2011

Academic Editor: Mark A. Petersen

Copyright © 2011 L. C. Jang. 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 consider the weighted -Genocchi numbers and polynomials. From the construction of the weighted -Genocchi numbers and polynomials, we investigate many interesting identities and relations satisfied by these new numbers and polynomials.

1. Introduction

Let be a fixed odd prime number. Throughout this paper, , , , and , will, respectively, denote the ring of -adic integers, the field, of -adic rational numbers, the complex number field and the completion of algebraic closure of . Let be the normalized exponential valuation of such that (see [116]).

As well-known definition, the Euler numbers and Genocchi numbers are defined by with the usual convention of replacing by and with the usual convention of replacing by . We assume that with and that the -number of is defined by (see [119]).

In [9], Kim introduced ordinary fermionic -adic integral on , and he studied some interesting relations and identities related to -extension of Euler numbers and polynomials. In [8], he also introduced the -extension of the ordinary fermionic -adic integral on and he investigated many physical properties related to -Euler numbers and polynomials. Recently, Kim firstly introduced the meaning of the weighted -Euler numbers and polynomials associated with the weighted -Bernstein polynomials by using the fermionic invariant -adic integral on (see [14, 15]). In [16], Ryoo tried to study the weighted -Euler number and polynomials by the same method of Kim et al. in [14] and the -extension of the fermionic -adic invariant integrals on . As well-known properties, the Genocchi numbers are integers. The first few Genocchi numbers for are . The first few prime Genocchi numbers are −3 and 17, which occur for and 8. There are no others with . These properties are very important to study in the area of fermionic distribution and -adic numbers theory. By this reason, many mathematicians and physicians have studied Genocchi and Euler numbers which are in the different areas. By the same motivation, we consider weighted -Genocchi polynomials and numbers by using the fermionic -adic -integral on which are constructed by Kim and Ryoo (cf. [8, 16]).

In this paper, we consider the -Genocchi numbers and polynomials with weighted . From the construction of the weighted -Genocchi numbers and polynomials, we investigate many interesting identities and relations satisfied by these new numbers and polynomials.

2. The Weighted -Genocchi Numbers and Polynomials

Let be the space of uniformly differentiable functions and, for , the fermionic -adic invariant integral of on is defined by Kim as follows: (see [116]). If we take , then we get By (1.2) and (2.2), we get From (2.3), For , the fermionic -adic -integral of on is defined by Kim as follows: (see [116]). From (2.5), we note that where and .

For , we consider the following fermionic -adic -integral on : where are called the th -Genocchi numbers with weight . From (2.7), we get By comparing the coefficients on the both sides of (2.7) and (2.8), we get From (2.9), we obtain the following theorem.

Theorem 2.1. For and , one has

By the definition of fermionic -adic -integrals, we get Therefore, we obtain the following theorem.

Theorem 2.2. For and , we have

By Theorem 2.2, we have the generating function of as follows: Let be the generating function of . Then, by (2.9) and (2.13), we get

The -Genocchi polynomials with weight are defined by From (2.15), we get By (2.15) and (2.16), we obtain the following theorem.

Theorem 2.3. For and , one has

We note that From (2.17) and (2.18), we obtain the following theorem.

Theorem 2.4. For and , one has

From (2.15), we note that Therefore, we obtain the following theorem.

Theorem 2.5. For , one has

From (2.15) and (2.21), we obtain that Therefore, we obtain the following theorem.

Theorem 2.6. For and , one has

From (2.6), if we take , then we get By (2.17) and (2.24), we obtain the following theorem.

Theorem 2.7. For , and , one has

We remark that if we take in Theorem 2.7, then we have and if we take in Theorem 2.7, then we have From (2.27) with , we obtain the following corollary.

Corollary 2.8. For and , one has

From (2.19), we note that From (2.29), we get with the usual convention about replacing by . By (2.28) and (2.30), we get From (2.28) and (2.31), we obtain the following theorem.

Theorem 2.9. For and , one has

Acknowledgment

This paper was supported by the Konkuk University in 2011.

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