# On the Generalized Open image in new window -Genocchi Numbers and Polynomials of Higher-Order

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## Abstract

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

## Keywords

Differential Equation Partial Differential Equation Ordinary Differential Equation Functional Analysis Functional Equation## 1. Introduction

(see [1]). In the special case Open image in new window , Open image in new window are called the Open image in new window th generalized Genocchi numbers attached to Open image in new window (see [1, 4, 5, 6]).

(see [7]). In the special case Open image in new window , Open image in new window are called the Open image in new window th generalized Genocchi numbers attached to Open image in new window of order Open image in new window (see [1, 4, 5, 6, 7, 8, 9]). From (1.3) and (1.4), we derive Open image in new window .

There is an unexpected connection with Open image in new window -analysis and quantum groups, and thus with noncommutative geometry Open image in new window -analysis is a sort of Open image in new window -deformation of the ordinary analysis. Spherical functions on quantum groups are Open image in new window -special functions. Recently, many authors have studied the Open image in new window -extension in various areas (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]). Govil and Gupta [10] have introduced a new type of Open image in new window -integrated Meyer-König-Zeller-Durrmeyer operators, and their results are closely related to the study of Open image in new window -Bernstein polynomials and Open image in new window -Genocchi polynomials, which are treated in this paper. In this paper, we first consider the Open image in new window -extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to Open image in new window . The purpose of this paper is to present a systemic study of some families of higher-order generalized Open image in new window -Genocchi numbers and polynomials attached to Open image in new window by using the generating function of those numbers and polynomials.

## 2. Generalized Open image in new window -Genocchi Numbers and Polynomials

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

Theorem 2.1.

Thus we obtain the following corollary.

Corollary 2.2.

where Open image in new window .

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 Open image in new window in Theorem 2.3, we obtain the following corollary.

Corollary 2.6.

Therefore, we obtain the following theorem.

Theorem 2.7.

## References

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