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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