Research Article | Open Access

Seog-Hoon Rim, Joohee Jeong, "Some New Identities on the -Genocchi Numbers and Polynomials with Weight
", *International Scholarly Research Notices*, vol. 2011, Article ID 271784, 6 pages, 2011. https://doi.org/10.5402/2011/271784

# Some New Identities on the -Genocchi Numbers and Polynomials with Weight

**Academic Editor:**A. Bellouquid

#### Abstract

We construct a new type of -Genocchi numbers and polynomials with weight . From these -Genocchi numbers and polynomials with weight , we establish some interesting identities and relations.

#### 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, and the completion of the algebraic closure of . Let be the set of natural numbers and . Let be the normalized exponential valuation of with . When one talks of -extension, is variously considered as an indeterminate, a complex , or a -adic number . In this paper, we assume that with . As a definition of -numbers, we use the notation of -number of (cf. [1β11]). Note that . Let be the space of continuous functions on . For , the -adic invariant integral on is defined by Kim [1, 3], From (1.2), we have the well-known integral equation (see [1, 3]), where , .

For , in [11], the -Genocchi polynomials with weight are introduced by By comparing the coefficients of both sides of (1.4), we have

In the special case, , are called the th -Genocchi numbers with weight .

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

In this section, we show some new identities on the -Genocchi numbers and polynomials with weight . And we establish the distribution relation for -Genocchi polynomials with weight .

From (1.5), we can easily see that From (1.5) and (2.1), we note that Note that and from (2.2), we have the relation of polynomials and numbers, with the usual convention of replacing by .

Thus, by (2.3), we have a theorem.

Theorem 2.1. *For and , one has
**In (1.3), if we take ,
**
We apply with (1.5), and we have the following:
**
By comparing the coefficients on both the sides in (2.6), we get
**
From (2.2) and (2.7), we can derive the following:
**
with the usual convention of replacing by . **For a fixed odd positive integer with , we set
**
where satisfies the condition . For the distribution relation for the -Genocchi polynomials with weight , we consider the following:
**
By (1.5) and (2.10), we get a theorem.*

Theorem 2.2. *For and , with , one has
*

#### 3. Higher-Order -Genocchi Numbers and Polynomials with Weight

In this section, we define higher-order -Genocchi polynomials and numbers with weight . We find an integral equation for higher-order -Genocchi numbers with weight . And we establish a combination property.

Let and , for , then we define higher-order -Genocchi polynomials with weight as follows: where .

In the special case, , are called the th -Genocchi numbers with weight .

In (3.1), apply the following identity: and we have a theorem.

Theorem 3.1. *For and , one has
*

We consider, for and , Therefore, we obtain the following combinatorial property.

Theorem 3.2. *For and , one has
*

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#### Copyright

Copyright © 2011 Seog-Hoon Rim and Joohee Jeong. 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.