1. Introduction
The Genocchi numbers and polynomials possess many interesting properties and are arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of the q-Genocchi numbers and polynomials (see [1–13]). In this paper, we construct a new type of q-Genocchi numbers and polynomials with weight α and weak weight β.
Throughout this paper, we use the following notations. By
ℤp, we denote the ring of
p-adic rational integers,
ℚp denotes the field of
p-adic rational numbers,
ℂp denotes the completion of algebraic closure of
ℚp,
ℕ denotes the set of natural numbers,
ℤ denotes the ring of rational integers,
ℚ denotes the field of rational numbers,
ℂ denotes the set of complex numbers, and
ℤ+ =
ℕ ∪ {0}. Let
νp be the normalized exponential valuation of
ℂp with
. When one talks of
q-extension,
q is considered in many ways such as an indeterminate, a complex number
q ∈
ℂ, or
p-adic number
q ∈
ℂp. If
q ∈
ℂ, one normally assume that |
q | < 1. If
q ∈
ℂp, we normally assumes that |
q − 1|
p <
p−(1/p−1) so that
qx = exp (
xlog
q) for |
x|
p ≤ 1. Throughout this paper, we use the notation
()
cf. [
1–
13].
Hence, lim
q→1[
x] =
x for any
x with |
x|
p ≤ 1 in the present
p-adic case. For
()
the fermionic
p-adic
q-integral on
ℤp is defined by Kim as follows:
()
cf. [
3–
6].
If we take
f1(
x) =
f(
x + 1) in (
1.1), then we easily see that
()
From (
1.4), we obtain
()
where
fn(
x) =
f(
x +
n) (cf. [
3–
6]).
As-well-known definition, the Genocchi polynomials are defined by
()
with the usual convention of replacing
Gn(
x) by
Gn(
x). In the special case,
x = 0,
Gn(0) =
Gn are called the
n-th Genocchi numbers (cf. [
1–
11]).
These numbers and polynomials are interpolated by the Genocchi zeta function and Hurwitz-type Genocchi zeta function, respectively.
()
Our aim in this paper is to define
q-Genocchi numbers
and polynomials
with weight
α and weak weight
β. We investigate some properties which are related to
q-Genocchi numbers
and polynomials
with weight
α and weak weight
β. We also derive the existence of a specific interpolation function which interpolates
q-Genocchi numbers
and polynomials
with weight
α and weak weight
β at negative integers.
2. q-Genocchi Numbers and Polynomials with Weight α and Weak Weight β
Our primary goal of this section is to define q-Genocchi numbers and polynomials with weight α and weak weight β. We also find generating functions of q-Genocchi numbers and polynomials with weight α and weak weight β.
For
α ∈
ℤ and
q ∈
ℂp with |1−
q|
p ≤ 1,
q-Genocchi numbers
are defined by
()
By using
p-adic
q-integral on
ℤp, we obtain
()
By (
2.1), we have
()
From the above, we can easily obtain that
()
Thus,
q-Genocchi numbers
with weight
α and weak weight
β are defined by means of the generating function
()
Using similar method as above, we introduce q-Genocchi polynomials with weight α and weak weight β.
are defined by
()
By using
p-adic
q-integral, we have
()
By using (
2.6) and (
2.7), we obtain
()
Remark 2.1. In (2.8), we simply see that
()
Since
, we easily obtain that
()
Observe that, if
q → 1, then
and
.
By (2.7), we have the following complement relation.
Theorem 2.2. Property of complement
()
By (2.7), we have the following distribution relation.
Theorem 2.3. For any positive integer m (=odd), one has
()
By (
1.5), (
2.1), and (
2.6), one easily sees that
()
Hence, we have the following theorem.
Theorem 2.4. Let m ∈ ℤ+.
If n ≡ 0 (mod 2), then
()
If
n ≡ 1 (mod 2), then
()
From (
1.4), one notes that
()
Therefore, we obtain the following theorem.
Theorem 2.5. For n ∈ ℤ+, one has
()
By Theorem 2.4 and (2.10), we have the following corollary.
Corollary 2.6. For n ∈ ℤ+, one has
()
with the usual convention of replacing
by
.
3. The Analogue of the Genocchi Zeta Function
By using
q-Genocchi numbers and polynomials with weight
α and weak weight
β,
q-Genocchi zeta function and Hurwitz
q-Genocchi zeta functions are defined. These functions interpolate the
q-Genocchi numbers and
q-Genocchi polynomials with weight
α and weak weight
β, respectively. In this section, we assume that
q ∈
ℂ with |
q | < 1. From (
2.4), we note that
()
By using the above equation, we are now ready to define
q-Genocchi zeta functions.
Definition 3.1. Let s ∈ ℂ. We define
()
Note that is a meromorphic function on ℂ. Note that, if q → 1, then which is the Genocchi zeta functions. Relation between and is given by the following theorem.
Theorem 3.2. For k ∈ ℕ, we have
()
Observe that
function interpolates
numbers at nonnegative integers. By using (
2.3), one notes that
()
()
By (3.2) and (3.5), we are now ready to define the Hurwitz q-Genocchi zeta functions.
Definition 3.3. Let s ∈ ℂ. We define
()
Note that
is a meromorphic function on
ℂ.
Remark 3.4. It holds that
()
Relation between and is given by the following theorem.
Theorem 3.5. For k ∈ ℕ, one has
()
Observe that
function interpolates
numbers at nonnegative integers.
