- CS Ryoo
^{1}, - T Kim
^{2}Email author, - J Choi
^{2}and - B Lee
^{3}

**2011**:424809

https://doi.org/10.1155/2011/424809

© C. S. Ryoo et al. 2011

**Received: **10 August 2010

**Accepted: **18 February 2011

**Published: **10 March 2011

## Abstract

We first consider the -extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to . The purpose of this paper is to present a systemic study of some families of higher-order generalized -Genocchi numbers and polynomials attached to by using the generating function of those numbers and polynomials.

## 1. Introduction

(see [1]). In the special case , are called the th generalized Genocchi numbers attached to (see [1, 4–6]).

(see [7]). In the special case , are called the th generalized Genocchi numbers attached to of order (see [1, 4–9]). From (1.3) and (1.4), we derive .

There is an unexpected connection with -analysis and quantum groups, and thus with noncommutative geometry -analysis is a sort of -deformation of the ordinary analysis. Spherical functions on quantum groups are -special functions. Recently, many authors have studied the -extension in various areas (see [1–15]). Govil and Gupta [10] have introduced a new type of -integrated Meyer-König-Zeller-Durrmeyer operators, and their results are closely related to the study of -Bernstein polynomials and -Genocchi polynomials, which are treated in this paper. In this paper, we first consider the -extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to . The purpose of this paper is to present a systemic study of some families of higher-order generalized -Genocchi numbers and polynomials attached to by using the generating function of those numbers and polynomials.

## 2. Generalized -Genocchi Numbers and Polynomials

In the special case , are called the th generalized -Genocchi numbers of order attached to . Therefore, we obtain the following theorem.

Theorem 2.1.

Thus we obtain the following corollary.

Corollary 2.2.

Therefore, we obtain the following theorem.

Theorem 2.3.

By (2.10) and (2.11), we obtain the following corollary.

Corollary 2.4.

By (2.7), we can derive the following corollary.

Corollary 2.5.

For in Theorem 2.3, we obtain the following corollary.

Corollary 2.6.

Therefore, we obtain the following theorem.

Theorem 2.7.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.