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