Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 860921, 9 pages
doi:10.1155/2012/860921
Research Article
Some Relations of the Twisted 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 12 February 2012; Accepted 9 March 2012
Academic Editor: Natig Atakishiyev
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.
Recently many mathematicians are working on Genocchi polynomials and Genocchi numbers. We
define a new type of twisted q-Genocchi numbers and polynomials with weight α and weak weight
β and give some interesting relations of the twisted q-Genocchi numbers and polynomials with
weight α and weak weight β. Finally, we find relations between twisted q-Genocchi zeta function
and twisted Hurwitz q-Genocchi zeta function.
1. Introduction
The Genocchi numbers and polynomials possess many interesting properties and arising in
many areas of mathematics and physics. Recently, many mathematicians have studied in the
area of the q-Genocchi numbers and polynomials see 1–16.
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 assume that |q − 1|p < p−1/p−1 so that
qx expx log q for |x|p ≤ 1. Throughout this paper we use the following notation:
2
xq 1 − qx
,
1−q
x−q
Abstract and Applied Analysis
x
1 − −q
1.1
1q
cf. 1–13.
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. 11–14.
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. 5–9.
n
Let Cpn {w | wp 1} be the cyclic group of order pn and let
Tp lim Cpn Cp∞ ∪n≥0 Cpn
n→∞
1.6
be the locally constant space. For w ∈ Tp , we denote by φw : Zp → Cp the locally constant
function x → wx .
As well-known definition, the Genocchi polynomials are defined by
∞
tn
2t
Gt
,
e
G
n
n!
et 1
n0
∞
tn
2t xt
Ft, x t
e eGxt Gn x ,
n!
e 1
n0
Ft 1.7
with the usual convention of replacing Gn x by Gn x. Gn 0 Gn are called the nth
Genocchi numbers cf. 2–5, 14.
Abstract and Applied Analysis
3
These numbers and polynomials are interpolated by the Genocchi zeta function and
Hurwitz-type Genocchi zeta function, respectively:
∞
−1n
,
ζG s 2
ns
n1
1.8
∞
−1n
ζG s, x 2
s.
n0 n x
α,β
α,β
Our aim in this paper is to define twisted q-Genocchi numbers Gn,q,w and polynomials
Gn,q,w x with weight α and weak weight β. We investigate some properties which are related
α,β
α,β
to Gn,q,w and Gn,q,w x. We also derive the existence of a specific interpolation function which
α,β
α,β
interpolate Gn,q,w and Gn,q,w x at negative integers.
2. Generating Functions of Twisted q-Genocchi Numbers and
Polynomials with Weight α and Weak Weight β
α,β
Our primary goal of this section is to define twisted q-Genocchi numbers Gn,q,w and
α,β
polynomials Gn,q,w x with weight α and weak weight β. We also find generating functions
α,β
α,β
of Gn,q,w and Gn,q,w x.
Definition 2.1. For α, β ∈ Q and q ∈ Cp with |1 − q|p ≤ 1,
α,β
Gn,q,w
n
Zp
φw xxn−1
qα dμ−qβ x.
2.1
α,β
We call Gn,q,w twisted q-Genocchi numbers with weight α and weak weight β.
By using p-adic q-integral on Zp , we obtain
n
p
−1
x
1
n−1
x
β
β
−q
φw xxn−1
dμ
w
n
lim
x
x
α
α
−q
q
q
N → ∞ pN
Zp
−qβ x0
N
n2qβ
∞
m βm
−1 q
w
m
2.2
mn−1
qα .
m0
From 2.1 and 2.2, we have
α,β
Gn,q,w
n2qβ
1
1 − qα
n−1 n−1 l0
1
n−1
.
−1l
l
1 wqβαl
2.3
We set
α,β
Fq,w t ∞
n0
α,β
Gn,q,w
tn
.
n!
2.4
4
Abstract and Applied Analysis
By using the previous equation and 2.3, we have
α,β
Fq,w t
∞
n2qβ
n0
∞
2qβ t
1
1 − qα
n−1 n−1 l0
tn
1
n−1
l
−1
l
1 wqβαl n!
2.5
−1m qβm wm emqα t .
m0
α,β
Thus twisted q-Genocchi numbers Gn,q,w with weight α and weak weight β are defined by
means of the generating function:
α,β
Fq,w t 2qβ t
∞
−1n qβn wn enqα t .
2.6
n0
By using 2.2, we have
∞
α,β
Gn,q,w
n0
tn
t
n!
Zp
φw xexqα t dμ−qβ x.
2.7
From 2.5 and 2.7, we have
∞
∞
n
α,β t
Gn,q,w 2qβ t −1m qβm wm emqα t .
n!
n0
m0
2.8
α,β
Next, we introduce twisted q-Genocchi polynomials Gn,q,w x with weight α and weak
weight β.
Definition 2.2. For α, β ∈ Q and q ∈ Cp with |1 − q|p ≤ 1,
α,β
Gn,q,w x
n
Zp
n−1
φw y x y qα dμ−qβ y .
2.9
α,β
We call Gn,q,w x twisted q-Genocchi polynomials with weight α and weak weight β.
By using p-adic q-integral, we have
p
−1
y
n−1
n−1 1
φw y x y qα dμ−qβ y n lim N wy x y qα −qβ
n
N →∞ p
Zp
−qβ y0
N
n2qβ
∞
m βm
−1 q
m0
w x m
mn−1
qα .
2.10
Abstract and Applied Analysis
5
By using 2.9 and 2.10, we obtain
α,β
Gn,q,w x n2qβ
1
1 − qα
n−1 n−1 1
n−1
.
−1l qαxl
l
1 wqβαl
l0
2.11
We set
α,β
Fq,w t, x ∞
α,β
Gn,q,w x
n0
tn
.
n!
2.12
By using the previous equation and 2.11, we have
α,β
Fq,w t, x
∞
n2qβ
n0
2qβ t
∞
1
1 − qα
m βm
−1 q
n−1 n−1 l0
tn
1
n−1
l αxl
−1 q
l
1 wqβαl n!
m xmqα t
w e
2.13
.
m0
α,β
Thus twisted q-Genocchi polynomials Gn,q,w x with weight α and weak weight β are
defined by means of the generating function:
α,β
Fq,w t, x 2qβ t
∞
−1m qβm wm exmqα t .
2.14
m0
By using 2.9, we have
∞
tn
α,β
Gn,q,w x t
φw y exyqα t dμ−qβ y .
n!
Zp
n0
2.15
By 2.13 and 2.15 we have
∞
∞
tn
α,β
Gn,q,w x 2qβ t −1m qβm wm exmqα t .
n!
n0
m0
2.16
Remark 2.3. In 2.14, we simply identify that
α,β
lim Fq,w t, x 2t
q→1
∞
−1n wn exnt
n0
2.17
Fw t, x.
α,β
α,β
Observe that if q → 1, then Fq,w t → Fw t and Fq,w t, x → Fw t, x. Note that if
α,β
α,β
q → 1 and w 1, then Gn,q,w → Gn and Gn,q,w x → Gn x.
6
Abstract and Applied Analysis
3. Some Relations between Twisted q-Genocchi Numbers and
Polynomials with Weight α and Weak Weight β
By 2.11, we have the following complement relation.
Theorem 3.1. One has the property of complement
α,β
α,β
Gn,q−1 ,w−1 1 − x −1n wqαn−1 Gn,q,w x.
3.1
Also, by 2.11, we have the following distribution relation.
Theorem 3.2. For any positive integer m(=odd), one has
α,β
Gn,q,w x
2qβ
2qβm
mn−1
qα
m−1
i
−1 w q
i βi
α,β
Gn,qm ,wm
i0
ix
,
m
n ∈ Z .
3.2
Let fx twx exqα t . Then by 1.5, left-hand side is in the following form:
∞ tm
α,β
α,β
qβn wn Gm,q,w n −1n−1 Gm,q,w
.
qβn I−qβ fn −1n−1 I−qβ f m!
m0
3.3
And right-hand side in 1.5 is in the following form:
2qβ
n−1
−1n−1−l qβl fl ∞
2qβ m
m0
l0
n−1
tm
.
−1n−1−l qβl wl lm−1
qα
m!
l0
3.4
By 3.3 and 3.4, one easily sees that
α,β
α,β
qβn wn Gm,q,w n −1n−1 Gm,q,w 2qβ m
n−1
−1n−1−l qβl wl lm−1
qα .
3.5
l0
Hence, we have the following theorem.
Theorem 3.3 Let m ∈ Z . If n ≡ 0 mod 2, then
α,β
α,β
qβn wn Gm,q,w n − Gm,q,w 2qβ m
n−1
−1l1 qβl wl lm−1
qα .
3.6
l0
If n ≡ 1 mod 2, then
α,β
α,β
qβn wn Gm,q,w n Gm,q,w 2qβ m
n−1
−1l qβl wl lm−1
qα .
l0
3.7
Abstract and Applied Analysis
7
Since x yqα xqα qαx yqα , one easily obtains that
α,β
Gn1,q,w x
n 1
Zp
n
φw y x y qα dμ−qβ y
n 12qβ
∞
3.8
−1m qβm wm x mnqα .
m0
From 1.4, one notes that
2qβ t qβ
Zp
∞ n0
twx1 ex1qα t dμ−qβ x n
α,β
α,β t
.
qβ wGn,q,w 1 Gn,q,w
n!
Zp
twx exqα t dμ−qβ x
3.9
By using comparing coefficients of tn /n! in the previous equation, we easily obtain the
following theorem.
Theorem 3.4. For n ∈ Z , one has
q
β
α,β
wGn,q,w 1
α,β
Gn,q,w
2qβ ,
0,
if n 1,
3.10
if n /
1.
By 3.8 and 3.10, we have the following corollary.
Corollary 3.5. For n ∈ Z , one has
q
β−α
w q
α
α,β
Gq,w
n
α,β
1 Gn,q,w α,β n
2qβ ,
0,
if n 1,
3.11
if n /
1,
α,β
with the usual convention of replacing Gq,w by Gn,q,w .
4. 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 2.5, we note that
∞
dk1 α,β α,β
F
t
12
k
−1m qβm wm mkqα Gk1,q,w
β
q,w
q
dtk1
m0
t0
k ∈ N.
By using the previous equation, we are now ready to define q-Genocchi zeta functions.
4.1
8
Abstract and Applied Analysis
Definition 4.1. Let s ∈ C. One has
α,β
ζq,w s 2qβ
∞
−1n qβn wn
nsqα
n1
4.2
.
α,β
Note that ζq,w s is a meromorphic function on C. Observe that if q
α,β
limq → 1 ζq,w s
→
1, then
ζw s.
α,β
α,β
Theorem 4.2. Relation between ζq,w s and Gk,q,w is given by
α,β
α,β
ζq,w −k
α,β
Gk1,q,w
k1
4.3
.
α,β
Observe that ζq,w s interpolates Gk,q,w at nonnegative integers.
By using 2.14, one notes that
∞
dk1 α,β
α,β
F
t,
x
12
k
−1m qβm wm x mkqα Gk1,q,w x,
β
q,w
q
k1
dt
m0
t0
k1 ∞
tn d
α,β
α,β
Gn,q,w x
Gk1,q,w x, for k ∈ N.
dt
n!
n0
4.4
4.5
t0
By 4.5, we are now ready to define the twisted Hurwitz q-Genocchi zeta functions.
Definition 4.3. Let s ∈ C. One has
α,β
ζq,w s, x 2qβ
∞
−1n qβn wn
n0
x nsqα
α,β
Note that ζq,w s, x is a meromorphic function on C. Observe that if q
α,β
limq → 1 ζq,w s, x
4.6
.
→
1, then
ζw s, x.
α
α
Theorem 4.4. Relation between ζq,w s, x and Gk,q,w x is given by
α,β
α,β
ζq,w −k, x
α,β
α,β
Gk1,q,w x
k1
.
Observe that ζq,w −k, x interpolates Gk,q,w x at nonnegative integers.
4.7
Abstract and Applied Analysis
9
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