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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 860921, 9 pages
http://dx.doi.org/10.1155/2012/860921
Research Article

Some Relations of the Twisted q-Genocchi Numbers and Polynomials with Weight α and Weak Weight β

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

Received 12 February 2012; Accepted 9 March 2012

Academic Editor: Natig Atakishiyev

Copyright © 2012 J. Y. Kang 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 many mathematicians are working on Genocchi polynomials and Genocchi numbers. We define a new type of twisted q-Genocchi numbers and polynomials with weight and weak weight and give some interesting relations of the twisted q-Genocchi numbers and polynomials with weight and weak weight . Finally, we find relations between twisted q-Genocchi zeta function and twisted Hurwitz q-Genocchi zeta function.

1. Introduction

The Genocchi numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of the -Genocchi numbers and polynomials (see [116]).

Throughout this paper we use the following notations. By we denote the ring of -adic rational integers, denotes the field of -adic rational numbers, denotes the completion of algebraic closure of , denotes the set of natural numbers, denotes the ring of rational integers, denotes the field of rational numbers, denotes the set of complex numbers, and . Let be the normalized exponential valuation of with . When one talks of -extension, is considered in many ways such as an indeterminate, a complex number , or -adic number . If , one normally assume that . If , we normally assume that so that for . Throughout this paper we use the following notation: (cf. [113]).

Hence, for any with in the present -adic case.

For the fermionic -adic -integral on is defined by Kim as follows: (cf. [1114]).

If we take in (1.1), then we easily see that From (1.4), we obtain where (cf. [59]).

Let be the cyclic group of order and let be the locally constant space. For , we denote by the locally constant function .

As well-known definition, the Genocchi polynomials are defined by with the usual convention of replacing by . are called the th Genocchi numbers (cf. [25, 14]).

These numbers and polynomials are interpolated by the Genocchi zeta function and Hurwitz-type Genocchi zeta function, respectively:

Our aim in this paper is to define twisted -Genocchi numbers and polynomials with weight and weak weight . We investigate some properties which are related to and . We also derive the existence of a specific interpolation function which interpolate and at negative integers.

2. Generating Functions of Twisted -Genocchi Numbers and Polynomials with Weight and Weak Weight

Our primary goal of this section is to define twisted -Genocchi numbers and polynomials with weight and weak weight . We also find generating functions of and .

Definition 2.1. For and with , We call twisted -Genocchi numbers with weight and weak weight .

By using -adic -integral on , we obtain From (2.1) and (2.2), we have

We set

By using the previous equation and (2.3), we have Thus twisted -Genocchi numbers with weight and weak weight are defined by means of the generating function:

By using (2.2), we have

From (2.5) and (2.7), we have

Next, we introduce twisted -Genocchi polynomials with weight and weak weight .

Definition 2.2. For and with , We call twisted -Genocchi polynomials with weight and weak weight .

By using -adic -integral, we have

By using (2.9) and (2.10), we obtain

We set

By using the previous equation and (2.11), we have

Thus twisted -Genocchi polynomials with weight and weak weight are defined by means of the generating function:

By using (2.9), we have

By (2.13) and (2.15) we have

Remark 2.3. In (2.14), we simply identify that
Observe that if , then and . Note that if and , then and .

3. Some Relations between Twisted -Genocchi Numbers and Polynomials with Weight and Weak Weight

By (2.11), we have the following complement relation.

Theorem 3.1. One has the property of complement Also, by (2.11), we have the following distribution relation.

Theorem 3.2. For any positive integer (=odd), one has Let . Then by (1.5), left-hand side is in the following form: And right-hand side in (1.5) is in the following form: By (3.3) and (3.4), one easily sees that Hence, we have the following theorem.

Theorem 3.3 (Let ). If , then
If , then
Since , one easily obtains that
From (1.4), one notes that

By using comparing coefficients of in the previous equation, we easily obtain the following theorem.

Theorem 3.4. For , one has By (3.8) and (3.10), we have the following corollary.

Corollary 3.5. For , one has with the usual convention of replacing by .

4. The Analogue of the Genocchi Zeta Function

By using -Genocchi numbers and polynomials with weight and weak weight , -Genocchi zeta function and Hurwitz -Genocchi zeta functions are defined. These functions interpolate the -Genocchi numbers and -Genocchi polynomials with weight and weak weight , respectively. In this section we assume that with .

From (2.5), we note that By using the previous equation, we are now ready to define -Genocchi zeta functions.

Definition 4.1. Let . One has Note that is a meromorphic function on . Observe that if , then .

Theorem 4.2. Relation between and is given by Observe that interpolates at nonnegative integers.

By using (2.14), one notes that By (4.5), we are now ready to define the twisted Hurwitz -Genocchi zeta functions.

Definition 4.3. Let . One has Note that is a meromorphic function on . Observe that if , then .

Theorem 4.4. Relation between and is given by Observe that interpolates at nonnegative integers.

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