Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 743939, 8 pages
doi:10.1155/2012/743939
Research Article
On the q-Genocchi Numbers and Polynomials with
Weight α and Weak Weight β
J. Y. Kang, H. Y. Lee, and N. S. Jung
Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea
Correspondence should be addressed to J. Y. Kang, rkdwjddnr2002@yahoo.co.kr
Received 20 November 2011; Accepted 19 February 2012
Academic Editor: Francis T. K. Au
Copyright q 2012 J. Y. Kang et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We construct a new type of q-Genocchi numbers and polynomials with weight α and weak weight
α,β
α,β
β : Gn,q , Gn,q x, respectively. Some interesting results and relationships are obtained.
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
α,β
α,β
a new type of q-Genocchi numbers Gn,q and polynomials Gn,q x 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 |p|p p−νp p p−1 . 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 expx log q for |x|p ≤ 1. Throughout this paper, we use the notation
x
1 − −q
1 − qx
1.1
,
,
x−q xq 1−q
1q
cf. 1–13.
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Journal of Applied Mathematics
Hence, limq → 1 x x for any x with |x|p ≤ 1 in the present p-adic case. For
f ∈ UD Zp f | f : Zp −→ Cp is uniformly differentiable function ,
1.2
the fermionic p-adic q-integral on Zp is defined by Kim as follows:
I−q f −1
p
x
1
fxdμ−q x lim N fx −q ,
N →∞ p
Zp
−q x0
N
1.3
cf. 3–6.
If we take f1 x fx 1 in 1.1, then we easily see that
qI−q f1 I−q f 2q f0.
1.4
n−1
qn I−q fn −1n−1 I−q f 2q −1n−1−l ql fl,
1.5
From 1.4, we obtain
l0
where fn x fx n cf. 3–6.
As-well-known definition, the Genocchi polynomials are defined by
Ft ∞
tn
2t
Gt
e
Gn ,
t
n!
e 1
n0
1.6
∞
2t xt
tn
Ft, x t
e eGxt Gn x .
n!
e 1
n0
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.
∞
−1n
,
ζG s 2
ns
n1
1.7
∞
−1n
ζG s, x 2
s.
n0 n x
α,β
α,β
Our aim in this paper is to define q-Genocchi numbers Gn,q and polynomials Gn,q x with
weight α and weak weight β. We investigate some properties which are related to q-Genocchi
α,β
α,β
numbers Gn,q and polynomials Gn,q x with weight α and weak weight β. We also derive
the existence of a specific interpolation function which interpolates q-Genocchi numbers
α,β
α,β
Gn,q and polynomials Gn,q x with weight α and weak weight β at negative integers.
Journal of Applied Mathematics
3
2. q-Genocchi Numbers and Polynomials with
Weight α and Weak Weight β
α,β
Our primary goal of this section is to define q-Genocchi numbers Gn,q
α,β
Gn,q x
and polynomials
with weight α and weak weight β. We also find generating functions of q-Genocchi
α,β
α,β
numbers Gn,q and polynomials Gn,q x with weight α and weak weight β.
α,β
For α ∈ Z and q ∈ Cp with |1 − q|p ≤ 1, q-Genocchi numbers Gn,q are defined by
α,β
Gn,q
n
Zp
xn−1
qα dμ−qβ x.
2.1
By using p-adic q-integral on Zp , we obtain
n
p
−1
x
1
β
−q
n lim N xn−1
α
q
N →∞ p
−qβ x0
N
Zp
xn−1
qα dμ−qβ x
n2qβ
n2qβ
1
1 − qα
∞
n−1 n−1 n − 1
1
−1l
1 qαlβ
l
l0
2.2
−1m qβm mn−1
qα .
m0
By 2.1, we have
α,β
Gn,q n2qβ
∞
−1m qβm mn−1
qα .
2.3
m0
From the above, we can easily obtain that
α,β
Fq
t ∞
n
α,β t
Gn,q
n!
n0
t2qβ
∞
2.4
m βm mqα t
−1 q
e
.
m0
α,β
Thus, q-Genocchi numbers Gn,q with weight α and weak weight β are defined by means of
the generating function
α,β
Fq
t t2qβ
∞
−1m qβm emqα t .
2.5
m0
α,β
Using similar method as above, we introduce q-Genocchi polynomials Gn,q x with
weight α and weak weight β.
4
Journal of Applied Mathematics
α,β
Gn,q x are defined by
α,β
Gn,q x n
Zp
n−1
xy
qα
dμ−qβ y .
2.6
By using p-adic q-integral, we have
α,β
Gn,q x
n2qβ
1
1 − qα
n−1 n−1 n − 1
l
l0
−1l qαxl
1
.
1 qαlβ
2.7
By using 2.6 and 2.7, we obtain
α,β
Fq
t, x ∞
∞
tn
α,β
Gn,q x t2qβ −1m qβm emxqα t .
n!
n0
m0
2.8
Remark 2.1. In 2.8, we simply see that
α,β
lim Fq
q→1
t, x 2t
∞
−1m emxt
m0
2.9
2t xt
e
1 et
Ft, x.
Since x yqα xqα qαx yqα , we easily obtain that
α,β
Gn1,q x
n 1
Zp
n
x y qα dμ−qβ y
q
−αx
n1 n 1
k0
q−αx
α,β
qαxk Gk,q
xn1−k
qα
k
α,β n1
xqα qαx Gq
n 12qβ
∞
2.10
−1m qβm x mnqα .
m0
α,β
α,β
Observe that, if q → 1, then Gn,q → Gn and Gn,q x → Gn x.
By 2.7, we have the following complement relation.
Theorem 2.2. Property of complement
α,β
α,β
Gn,q−1 1 − x −1n−1 qαn−1 Gn,q x.
By 2.7, we have the following distribution relation.
2.11
Journal of Applied Mathematics
5
Theorem 2.3. For any positive integer m (odd), one has
2qβ
α,β
Gn,q x 2qβm
mn−1
qα
m−1
α,β
−1i qβi Gn,qm
i0
ix
,
m
n ∈ Z .
2.12
By 1.5, 2.1, and 2.6, one easily sees that
m2qβ
n−1
n−1 α,β
βn α,β
Gm,q .
−1n−1−l qβl lm−1
qα q Gm,q n −1
2.13
l0
Hence, we have the following theorem.
Theorem 2.4. Let m ∈ Z .
If n ≡ 0 mod 2, then
α,β
α,β
qβn Gm,q n − Gm,q m2qβ
n−1
−1l1 qβl lm−1
qα .
2.14
l0
If n ≡ 1 mod 2, then
α,β
α,β
qβn Gm,q n Gm,q m2qβ
n−1
−1l qβl lm−1
qα .
2.15
l0
From 1.4, one notes that
2qβ t qβ
Zp
∞
∞ Zp
texqα t dμ−qβ x
q
n0
tex1qα t dμ−qβ x β
Zp
nx α,β
1n−1
qα dμ−qβ x
α,β
qβ Gn,q 1 Gn,q
tn
n0
n!
Zp
nxn−1
qα dμ−qβ x
tn
n!
2.16
.
Therefore, we obtain the following theorem.
Theorem 2.5. For n ∈ Z , one has
α,β
α,β
qβ Gn,q 1 Gn,q ⎧
⎨2qβ ,
if n 1,
⎩0,
if n /
1.
By Theorem 2.4 and 2.10, we have the following corollary.
2.17
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Journal of Applied Mathematics
Corollary 2.6. For n ∈ Z , one has
⎧
⎨2qβ ,
n
α,β
1 Gn,q ⎩0,
α,β
qβ−α qα Gq
α,β n
with the usual convention of replacing Gq
if n 1,
2.18
if n /
1.
α,β
by Gn,q .
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
∞
dk1 α,β F
12
k
t
−1m qβm mkqα
β
q
q
dtk1
m0
t0
α,β
Gk1,q ,
3.1
k ∈ N.
By using the above equation, we are now ready to define q-Genocchi zeta functions.
Definition 3.1. Let s ∈ C. We define
α,β
ζq
s 2qβ
∞
−1n qβn
n1
α,β
Note that ζq
nsqα
3.2
.
α,β
s is a meromorphic function on C. Note that, if q → 1, then ζq
ζs which is the Genocchi zeta functions. Relation between
following theorem.
α,β
ζq s
and
α,β
Gk,q
s is given by the
Theorem 3.2. For k ∈ N, we have
α,β
α,β
ζq −k
Gk1,q
k1
.
3.3
Journal of Applied Mathematics
α,β
Observe that ζq
one notes that
7
α,β
s function interpolates Gk,q numbers at nonnegative integers. By using 2.3,
∞
dk1 α,β
F
12
x
k
t,
−1m qβm x mkqα
β
q
q
dtk1
m0
t0
d
dt
k1 ∞
n! t
α,β
Gn,q x
n0
α,β
k ∈ N,
α,β
for k ∈ N.
Gk1,q x,
n
Gk1,q x,
3.4
3.5
t0
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
α,β
ζq
α,β
Note that ζq
s, x 2qβ
∞
−1n qβn
.
3.6
∞
−1n
s, x 2
s.
n0 n x
3.7
n0 n
xsqα
s, x is a meromorphic function on C.
Remark 3.4. It holds that
α,β
lim ζq
q→1
α
α
Relation between ζq s, x and Gk,q x is given by the following theorem.
Theorem 3.5. For k ∈ N, one has
α,β
α,β
ζq −k, x
α,β
Observe that ζq
Gk1,q x
k1
.
3.8
α,β
−k, x function interpolates Gk,q x numbers at nonnegative integers.
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