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On the Generalized
-Genocchi Numbers and Polynomials of Higher-Order
Advances in Difference Equations volume 2011, Article number: 424809 (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
As a well known definition, the Genocchi polynomials are defined by

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

(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

(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

(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

The -factorial is defined by

and the Gaussian binomial coefficient is also defined by


It is known that

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

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:

Note that

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

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

Note that

Thus we obtain the following corollary.
Corollary 2.2.
For , we have

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

From (2.7), one notes that

where .
Therefore, we obtain the following theorem.
Theorem 2.3.
For , one has

Note that

By (2.10), one sees that

By (2.10) and (2.11), we obtain the following corollary.
Corollary 2.4.
For , we have

By (2.7), we can derive the following corollary.
Corollary 2.5.
For with
, we have

For in Theorem 2.3, we obtain the following corollary.
Corollary 2.6.
For , one has

In particular,

Let in Corollary 2.6. Then one has

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

By (2.17), one sees that

It is easy to see that

Therefore, we obtain the following theorem.
Theorem 2.7.
For , one has

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Ryoo, C., Kim, T., Choi, J. et al. On the Generalized -Genocchi Numbers and Polynomials of Higher-Order.
Adv Differ Equ 2011, 424809 (2011). https://doi.org/10.1155/2011/424809
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DOI: https://doi.org/10.1155/2011/424809
Keywords
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Functional Equation