On the Generalized -Genocchi Numbers and Polynomials of Higher-Order

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

As a well known definition, the Genocchi polynomials are defined by

(11)

where we use the technical method's notation by replacing by , symbolically, (see [1, 2]). In the special case , are called the th Genocchi numbers. From the definition of Genocchi numbers, we note that , and even coefficients are given by (see [3]), where is a Bernoulli number and is an Euler polynomial. The first few Genocchi numbers for are . The first few prime Genocchi numbers are given by and . It is known that there are no other prime Genocchi numbers with . For a real or complex parameter , the higher-order Genocchi polynomials are defined by

(12)

(see [1, 4]). In the special case , are called the th Genocchi numbers of order . From (1.1) and (1.2), we note that . For with , let be the Dirichlet character with conductor . It is known that the generalized Genocchi polynomials attached to are defined by

(13)

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

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

(14)

(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 .

Let us assume that with as an indeterminate. Then we, use the notation

(15)

The -factorial is defined by

(16)

and the Gaussian binomial coefficient is also defined by

(17)

(see [5, 10]). Note that

(18)

It is known that

(19)

(see [5, 10]). The -binomial formula are known that

(110)

(see[10, 11]).

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

For , let us consider the -extension of the generalized Genocchi polynomials of order attached to as follows:

(21)

Note that

(22)

By (2.1) and (1.4), we can see that . From (2.1), we note that

(23)

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

Theorem 2.1.

For , one has

(24)

Note that

(25)

Thus we obtain the following corollary.

Corollary 2.2.

For , we have

(26)

For and , one also considers the extended higher-order generalized -Genocchi polynomials as follows:

(27)

From (2.7), one notes that

(28)

where .

Therefore, we obtain the following theorem.

Theorem 2.3.

For , one has

(29)

Note that

(210)

By (2.10), one sees that

(211)

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

Corollary 2.4.

For , we have

(212)

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

Corollary 2.5.

For with , we have

(213)

For in Theorem 2.3, we obtain the following corollary.

Corollary 2.6.

For , one has

(214)

In particular,

(215)

Let in Corollary 2.6. Then one has

(216)

Let . Then, one has defines Barnes' type generalized -Genocchi polynomials attached to as follows:

(217)

By (2.17), one sees that

(218)

It is easy to see that

(219)

Therefore, we obtain the following theorem.

Theorem 2.7.

For , one has

(220)