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Journal of Applied Mathematics

VolumeΒ 2012Β (2012), Article IDΒ 743939, 8 pages

http://dx.doi.org/10.1155/2012/743939

## On the -Genocchi Numbers and Polynomials with Weight and Weak Weight

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

Received 20 November 2011; Accepted 19 February 2012

Academic Editor: Francis T. K.Β Au

Copyright Β© 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

We construct a new type of -Genocchi numbers and polynomials with weight and weak weight , respectively. Some interesting results and relationships are obtained.

#### 1. Introduction

The Genocchi numbers and polynomials possess many interesting properties and are arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of the -Genocchi numbers and polynomials (see [1β13]). In this paper, we construct a new type of -Genocchi numbers and polynomials with weight and weak weight .

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 assumes that so that for . Throughout this paper, we use the notation cf. [1β13].

Hence, for any with in the present -adic case. For the fermionic -adic -integral on is defined by Kim as follows: cf. [3β6].

If we take in (1.1), then we easily see that From (1.4), we obtain where (cf. [3β6]).

As-well-known definition, the Genocchi polynomials are defined by with the usual convention of replacing by . In the special case, , are called the -th Genocchi numbers (cf. [1β11]).

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 -Genocchi numbers and polynomials with weight and weak weight . We investigate some properties which are related to -Genocchi numbers and polynomials with weight and weak weight . We also derive the existence of a specific interpolation function which interpolates -Genocchi numbers and polynomials with weight and weak weight at negative integers.

#### 2. -Genocchi Numbers and Polynomials with Weight and Weak Weight

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

For and with , -Genocchi numbers are defined by By using -adic -integral on , we obtain By (2.1), we have From the above, we can easily obtain that Thus, -Genocchi numbers with weight and weak weight are defined by means of the generating function

Using similar method as above, we introduce -Genocchi polynomials with weight and weak weight .

are defined by By using -adic -integral, we have By using (2.6) and (2.7), we obtain

*Remark 2.1. *In (2.8), we simply see that
Since , we easily obtain that
Observe that, if , then and .

By (2.7), we have the following complement relation.

Theorem 2.2. *Property of complement
*

By (2.7), we have the following distribution relation.

Theorem 2.3. *For any positive integer (=odd), one has
**
By (1.5), (2.1), and (2.6), one easily sees that
*

Hence, we have the following theorem.

Theorem 2.4. *Let .**If , then
**
If , then
**
From (1.4), one notes that
*

Therefore, we obtain the following theorem.

Theorem 2.5. *For , one has
*

By Theorem 2.4 and (2.10), we have the following corollary.

Corollary 2.6. *For , one has
**
with the usual convention of replacing by . *

#### 3. 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 (2.4), we note that By using the above equation, we are now ready to define -Genocchi zeta functions.

*Definition 3.1. *Let . We define

Note that is a meromorphic function on . Note that, if , then which is the Genocchi zeta functions. Relation between and is given by the following theorem.

Theorem 3.2. *For , we have
**Observe that function interpolates numbers at nonnegative integers. By using (2.3), one notes that
*

By (3.2) and (3.5), we are now ready to define the Hurwitz -Genocchi zeta functions.

*Definition 3.3. *Let . We define
Note that is a meromorphic function on .

*Remark 3.4. *It holds that

Relation between and is given by the following theorem.

Theorem 3.5. *For , one has
**
Observe that function interpolates numbers at nonnegative integers.*

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