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 ), 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 ). 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 ). 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
(see [5, 10]). Note that
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  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.