Advances in Difference Equations

, 2011:424809 | Cite as

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

Open Access
Review Article

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

As a well known definition, the Genocchi polynomials are defined by
where we use the technical method's notation by replacing Open image in new window by Open image in new window , symbolically, (see [1, 2]). In the special case Open image in new window , Open image in new window are called the Open image in new window th Genocchi numbers. From the definition of Genocchi numbers, we note that Open image in new window , and even coefficients are given by Open image in new window (see [3]), where Open image in new window is a Bernoulli number and Open image in new window is an Euler polynomial. The first few Genocchi numbers for Open image in new window are Open image in new window . The first few prime Genocchi numbers are given by Open image in new window and Open image in new window . It is known that there are no other prime Genocchi numbers with Open image in new window . For a real or complex parameter Open image in new window , the higher-order Genocchi polynomials are defined by
(see [1, 4]). In the special case Open image in new window , Open image in new window are called the Open image in new window th Genocchi numbers of order Open image in new window . From (1.1) and (1.2), we note that Open image in new window . For Open image in new window with Open image in new window , let Open image in new window be the Dirichlet character with conductor Open image in new window . It is known that the generalized Genocchi polynomials attached to Open image in new window are defined by

(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]).

For a real or complex parameter Open image in new window , the generalized higher-order Genocchi polynomials attached to Open image in new window are also defined by

(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 .

Let us assume that Open image in new window with Open image in new window as an indeterminate. Then we, use the notation
The Open image in new window -factorial is defined by
and the Gaussian binomial coefficient is also defined by
(see [5, 10]). Note that
It is known that
(see [5, 10]). The Open image in new window -binomial formula are known that

(see[10, 11]).

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

For Open image in new window , let us consider the Open image in new window -extension of the generalized Genocchi polynomials of order Open image in new window attached to Open image in new window as follows:
By (2.1) and (1.4), we can see that Open image in new window . From (2.1), we note that

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.

For Open image in new window and Open image in new window , one also considers the extended higher-order generalized Open image in new window -Genocchi polynomials as follows:
From (2.7), one notes that

where Open image in new window .

Therefore, we obtain the following theorem.

Theorem 2.3.

Note that
By (2.10), one sees that

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.

In particular,
Let Open image in new window in Corollary 2.6. Then one has
Let Open image in new window . Then, one has defines Barnes' type generalized Open image in new window -Genocchi polynomials attached to Open image in new window as follows:
By (2.17), one sees that
It is easy to see that

Therefore, we obtain the following theorem.

Theorem 2.7.

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Copyright information

© C. S. Ryoo et al. 2011

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.

Authors and Affiliations

  1. 1.Department of MathematicsHannam UniversityDaejeonRepublic of Korea
  2. 2.Division of General Education-MathematicsKwangwoon UniversitySeoulRepublic of Korea
  3. 3.Department of Wireless Communications EngineeringKwangwoon UniversitySeoulRepublic of Korea

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