Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 487490, 7 pages
doi:10.1155/2011/487490
Research Article
New Construction Weighted
h, q-Genocchi Numbers and Polynomials
Related to Zeta Type Functions
Serkan Araci,1 Jong Jin Seo,2 and Dilek Erdal1
1
Department of Mathematics, Faculty of Arts and Science, University of Gaziantep,
27310 Gaziantep, Turkey
2
Department of Applied Mathematics, Pukyong National University,
Busan 608-737, Republic of Korea
Correspondence should be addressed to Jong Jin Seo, seo2011@pknu.ac.kr
Received 31 March 2011; Revised 17 June 2011; Accepted 11 July 2011
Academic Editor: Guang Zhang
Copyright q 2011 Serkan Araci 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.
The fundamental aim of this paper is to construct h, q-Genocchi numbers and polynomials with
weight α. We shall obtain some interesting relations by using p-adic q-integral on Zp in the sense of
fermionic. Also, we shall derive the h, q-extensions of zeta type functions with weight α from the
Mellin transformation of this generating function which interpolates the h, q-Genocchi numbers
and polynomials with weight α at negative integers.
1. Introduction, Definitions, and Notations
Let p be a fixed odd prime number. Throughout this paper we use the following notations.
Zp denotes the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp
denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic
closure of Qp . Let N be the set of natural numbers and N∗ N ∪ {0}. The p-adic absolute
value is defined by |p|p 1/p. In this paper, we assume |q − 1|p < 1 as an indeterminate. In
1–3, Kim defined the fermionic p-adic q-integral on Zp as follows:
I−q f −1
p
x
1
fxdμ−q x lim N fx −q .
N
→
∞
p −q x0
Zp
N
1.1
2
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xq is a q-extension of x which is defined by
xq 1 − qx
,
1−q
1.2
see 1–15.
Note that limq → 1 xq x.
Let fn x fx n. By the definition 1.1 we easily get
−1
p
x1
1
−qI−q f1 lim N fx 1 −q
N →∞ p
−q x0
N
N
N
−1
p
f pN qp f0
x 1
lim N fx −q − 1 q lim
N
N →∞ p
N →∞
1 qp
−q x0
1.3
I−q f − 2q f0.
Continuing this process, we obtain easily the relation
n−1
qn I−q fn −1n−1 I−q f 2q −1n−l−1 ql fl,
1.4
l0
h, q-Genocchi numbers are defined as follows:
h
G0,q 0,
n
h
qh−2 qGq 1
h
Gn,q h n
⎧
⎨2q , if n 1,
⎩0,
if n > 1,
1.5
h
with the usual convention about replacing Gq by Gn,q see 6.
In this paper, we constructed h, q-Genocchi numbers and polynomials with weight
α. By using fermionic p-adic q-integral equations on Zp , we investigated some interesting
identities and relations on the h, q-Genocchi numbers and polynomials with weight α.
Furthermore, we derive the q-extensions of zeta type functions with weight α from the
Mellin transformation of this generating function which interpolates the h, q-Genocchi
polynomials with weight α.
2. On the Weighted h, q-Genocchi Numbers and Polynomials
In this section, by using fermionic p-adic q-integral equations on Zp , some interesting
identities and relation on the h, q-Genocchi numbers and polynomials with weight α are
shown.
Definition 2.1. Let α, n ∈ N∗ and h ∈ N. Then the h, q-Genocchi numbers with weight α
defined by as follows:
α,h
Gn1,q
n1
2q
∞
−1m qmh mnqα .
m0
2.1
Discrete Dynamics in Nature and Society
3
α,1
α
If we take h 1 to 2.1, then we have, Gn1,q Gn1,q see 5.
From 2.1, we obtain
α,h
Gn1,q
n1
2q
1−q
α n
2q
1 − qα
2q
1 − qα
n
∞
m mh
−1 q
m0
n
−1
l
∞
l
l
l mα l
−1 q
l
2.2
−1m qmαlmh
m0
n
n
l0
n
n
l0
n
n
l0
n
qα
1−
n
−1m qmh 1 − qmα
m0
2q
∞
−1l
1
1 qαlh
Therefore, we obtain the following theorem.
Theorem 2.2. For α, n ∈ N∗ and h ∈ N. Then
α,h
Gn1,q
n1
2q
1−
n
qα
n
n
l0
l
−1l
1
.
1 qαlh
2.3
In 1.1, one takes fx qh−1x xnqα ,
Zp
q
h−1x
xnqα dμ−q x
1
1−q
α n
1
1−
n
qα
1
1 − qα
2q
1−
n
n
qα
n
n
l0
l
−1l
n
n
l0
N
−1
p
1
−qαlh
−1 lim N N →∞ p
−q x0
l
l
n
n
l0
−1
l
n
n
l0
l
Zp
qxαlh−1 dμ−q x
l
−1l
1q
lim
1 qαlh N → ∞
x
p N
1 qαlh
1 qp
N
2.4
1
1 qαlh
α,h
Gn1,q
n1
.
From 12, we obtain h, q-Genocchi numbers with weight α witt’s type formula as
follows.
4
Discrete Dynamics in Nature and Society
Theorem 2.3. For α, n ∈ N∗ and h ∈ N. Then
α,h
Gn1,q
n1
Zp
2.5
qh−1x xnqα dμ−q x.
From 2.1, one easily gets
Zp
qh−1x etxqα dμ−q x t2q
∞
−1m qmh etmqα .
2.6
m0
By 2.6, one has
∞
∞
n
α,h t
t2q −1m qmh etmqα .
Gn,q
n!
n0
m0
2.7
Therefore, we obtain the following corollary.
α,h
α,h
Corollary 2.4. If G0,q 0. Let Dq
α,h
Dq
t ∞
n0
t t2q
α,h
Gn,q tn /n!. Then
∞
−1m qmh etmqα .
2.8
m0
Now, one considers the h, q-Genocchi polynomials with weight α as follows:
α,h
Gn1,q x
n1
Zp
n
qh−1y x y qα dμ−q y ,
n ∈ N, α ∈ N∗ .
2.9
From 2.9, one sees that
α,h
Gn1,q x
n1
α,h
Let Dq
2q
t, x 1−q
∞
α n
n
n
l0
l
−1l qαlx
∞
1
2
−1m qmh m xnqα .
q
1 qαlh
m0
2.10
α,h
n0
Gn,q xtn /n!. Then, one has
α,h
Dq
t, x t2q
∞
m0
∞
tn
.
n!
2.11
−1n−l−1 qhl lm
qα .
2.12
−1m qmh etmxqα n0
α,h
Gn,q x
By 1.4, one sees that
α,h
q
hn
Gm1,q n
m1
α,h
−1
n−1
Gm1,q
m1
2q
n−1
l0
Discrete Dynamics in Nature and Society
5
Therefore, we obtain the following theorem.
Theorem 2.5. For m, h ∈ N, and α, n ∈ N∗ , one has
α,h
q
hn
Gm1,q n
m1
α,h
−1
n−1
Gm1,q
n−1
2q
m1
−1n−l−1 qhl lm
qα .
2.13
l0
In 1.3, it is known that
qI−q f1 I−q f 2q f0.
2.14
If we take fx qh−1x etxqα , then one has
2q q
Zp
∞
qh−1x1 etx1qα dμ−q x ⎛
⎝qh
m0
α,h
Gm1,q 1
m1
α,h
Gm1,q
Zp
qh−1x etxqα dμ−q x
⎞
2.15
m
⎠t .
m 1 m!
Therefore, by 2.15, we obtain the following theorem.
Theorem 2.6. For α ∈ N∗ and m, h ∈ N, one has
α,h
α,h
G0,q 0,
qh
Gm1,q 1
m1
α,h
Gm1,q
m1
⎧
⎨2q , if m 0,
⎩0,
if m /
0.
2.16
From 2.9, one can easily derive the following:
n d−1
x a
n
n
dqα y
qh−1y x y qα dμ−q y qdyh−1
dμ−qd y
−1a qha
dα
d
d−q a0
q
Zp
Zp
α,h
d−1
Gn1,qd x a/d
dnqα .
−1a qha
n1
d−q a0
2.17
Therefore, by 2.17, we obtain the following theorem.
Theorem 2.7. For d ≡ 1mod 2, n ∈ N∗ and α, h ∈ N
α,h
Gn1,q x d−1
dnqα d−q a0
α,h
−1a qha Gn1,qd
x a
d
.
2.18
6
Discrete Dynamics in Nature and Society
α,h
3. Interpolation Function of the Polynomials Gn,q x
In this section, we give interpolation function of the generating functions of h, q-Genocchi
polynomials with weight α. For s ∈ C and h ∈ N, by applying the Mellin transformation to
2.11, we obtain
α,h
Iq s, x
1
Γs
∞
t
0
s−2
α,h
−Dq −t, x
dt 2q
∞
m mh
−1 q
m0
1
Γs
∞
0
ts−1 e−tmxqα dt, 3.1
so we have
α,h
Iq
s, x 2q
∞
−1m qmh
m0 m
xsqα
.
3.2
.
3.3
We define q-extension zeta type function as follows.
Theorem 3.1. For s ∈ C, h ∈ N, and α ∈ N∗ . One has
α,h
Iq
α,h
Iq
s, x 2q
∞
−1m qmh
m0 m
xsqα
s, x can be continued analytically to an entire function.
By subsituting s −n into 3.3 one easily gets
α,h
α,h
Iq −n, x
Gn1,q x
n1
.
3.4
We obtain the following theorem.
Theorem 3.2. For h ∈ N and q, s ∈ C, |q| < 1. Then one defines
α,h
α,h
Iq −n, x
Gn1,q x
n1
.
3.5
References
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Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007.
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Physics, vol. 17, no. 2, pp. 218–225, 2010.
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fermionic p-adic integral on Zp ,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491,
2009.
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Discrete Dynamics in Nature and Society
7
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Their Interpolation function,” submitted.
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related integrals,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 179430, 9 pages, 2010.
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