Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 179385, 13 pages
doi:10.1155/2012/179385
Research Article
Some Properties of Multiple Generalized
q-Genocchi Polynomials with Weight α and
Weak Weight β
J. Y. Kang
Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea
Correspondence should be addressed to J. Y. Kang, rkdwjddnr2002@yahoo.co.kr
Received 28 May 2012; Revised 21 August 2012; Accepted 22 August 2012
Academic Editor: Cheon Ryoo
Copyright q 2012 J. Y. Kang. 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 present paper deals with the various q-Genocchi numbers and polynomials. We define a new
type of multiple generalized q-Genocchi numbers and polynomials with weight α and weak weight
β by applying the method of p-adic q-integral. We will find a link between their numbers and
polynomials with weight α and weak weight β. Also we will obtain the interesting properties
of their numbers and polynomials with weight α and weak weight β. Moreover, we construct a
Hurwitz-type zeta function which interpolates multiple generalized q-Genocchi polynomials with
weight α and weak weight β and find some combinatorial relations.
1. Introduction
Let p be a fixed odd prime number. Throughout this paper Zp , Qp , C, and Cp denote the ring
of p-adic rational integers, the field of p-adic rational numbers, the complex number field,
and the completion of the algebraic closure of Qp , respectively. Let N be the set of natural
numbers and Z N ∪ {0}. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p see 1–21. When one talks of q-extension, q is variously considered as an
indeterminate, a complex q ∈ C, or a p-adic number q ∈ Cp . If q ∈ C, then one normally
assumes |q| < 1. If q ∈ Cp , then we assume that |q − 1|p < 1.
Throughout this paper, we use the following notation:
1 − qx
,
xq 1−q
x−q
x
1 − −q
.
1q
Hence limq → 1 xq x for all x ∈ Zp see 1–14, 16, 18, 20, 21.
1.1
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We say that g : Zp → Cp is uniformly differentiable function at a point a ∈ Zp and we
write g ∈ UDZp if the difference quotients Φg : Zp × Zp → Cp such that
gx − g y
Φg x, y x−y
1.2
have a limit g a as x, y → a, a.
Let d be a fixed integer, and let p be a fixed prime number. For any positive integer N,
we set
X Xd lim
←
N
∗
X Z
,
X1 Zp ,
dpN Z
a dpZp ,
0<a<dp
a,p1
1.3
a dpN Zp x ∈ X | x ≡ a mod dpN ,
where a ∈ Z lies in 0 ≤ a < dpN .
For any positive integer N,
qa
μq a dpN Zp N dp q
1.4
is known to be a distribution on X.
For g ∈ UDZp , Kim defined the q-deformed fermionic p-adic integral on Zp :
I−q g −1
p
x
1
gxdμ−q x lim N gx −q .
N
→
∞
p −q x0
Zp
N
1.5
see 1–13, and note that
Zp
gxdμ−q x gxdμ−q x.
X
1.6
We consider the case q ∈ −1, 0 corresponding to q-deformed fermionic certain and
annihilation operators and the literature given there in 9, 13, 14.
In 9, 12, 14, 19, we introduced multiple generalized Genocchi number and
polynomials. Let χ be a primitive Dirichlet character of conductor f ∈ N. We assume that f
International Journal of Mathematics and Mathematical Sciences
3
r
is odd. Then the multiple generalized Genocchi numbers, Gn,χ , and the multiple generalized
r
Genocchi polynomials, Gn,χ x, associated with χ, are defined by
⎛
r
Fχ t
⎝
2t
⎛
Fχ t, x ⎝
r
r
2t
f−1
a0
χa−1a eat
eft 1
f−1
a0
⎞r
⎠ ∞
r
Gn,χ
n0
a at
χa−1 e
eft 1
⎞r
etx ⎠ tn
,
n!
1.7
∞
r
Gn,χ x
n0
n
t
.
n!
r
In the special case x 0, Gn,χ Gn,χ 0 are called the nth multiple generalized Genocchi
numbers attached to χ.
Now, having discussed the multiple generalized Genocchi numbers and polynomials,
we were ready to multiple-generalize them to their q-analogues. In generalizing the
generating functions of the Genocchi numbers and polynomials to their respective qanalogues; it is more useful than defining the generating function for the Genocchi numbers
and polynomials see 12.
α,β,r
Our aim in this paper is to define multiple generalized q-Genocchi numbers Gn,χ,q and
α,β,r
polynomials Gn,χ,q x with weight α and weak weight β. We investigate some properties
α,β,r
which are related to multiple generalized q-Genocchi numbers Gn,χ,q
α,β,r
Gn,χ,q x
and polynomials
with weight α and weak weight β. We also derive the existence of a specific
α,β,r
interpolation function which interpolate multiple generalized q-Genocchi numbers Gn,χ,q
α,β,r
and polynomials Gn,χ,q x with weight α and weak weight β at negative integers.
2. The Generating Functions of Multiple Generalized q-Genocchi
Numbers and Polynomials with Weight α and Weak Weight β
Many mathematicians constructed various kinds of generating functions of the q-Gnocchi
numbers and polynomials by using p-adic q-Vokenborn integral. First we introduce multiple
generalized q-Genocchi numbers and polynomials with weight α and weak weight β.
α,β
α,β
Let us define the generalized q-Genocchi numbers Gn,χ,q and polynomials Gn,χ,q x
with weight α and weak weight β, respectively,
α,β
Fχ,q t
α,β
Fχ,q t, x
∞
n
α,β t
Gn,χ,q tχxexqα t dμ−qβ x,
n!
X
n0
∞
tn
α,β
Gn,χ,q x tχ y exyqα t dμ−qβ y .
n!
X
n0
2.1
By using the Taylor expansion of exqα t , we have
∞ n0
X
t
χxxnqα dμ−qβ x
n
n!
∞
n0
α,β t
Gn,χ,q
n−1
n!
α,β
α,β
G0,χ,q
∞ G
n
n1,χ,q t
.
n 1 n!
n0
2.2
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By comparing the coefficient of both sides of tn /n! in 2.2, we get
α,β
Gn1,χ,q
2qβ
n1
1−q
α n
f−1
n 1
n
.
−1a qβa χa
−1l qαal
fαlβ
l
1
q
a0
l0
2.3
From 2.2 and 2.3, we can easily obtain that
n
∞
∞ ∞
n
t
α,β t
2qβ t −1l qβl χlelqα t .
Gn,χ,q χxxnqα dμ−qβ x
t
n! n0
n!
X
n0
l0
2.4
Therefore, we obtain
α,β
Fχ,q t 2qβ t
∞
∞
n
α,β t
Gn,χ,q .
−1l qβl χlelqα t n!
n0
l0
2.5
Similarly, we find the generating function of generalized q-Genocchi polynomials with
weight α and weak weight β:
α,β
α,β
G0,χ,q x
Gn1,χ,q x
0,
n1
X
∞
n
χ y x y qα dμ−qβ y 2qβ −1l qβl χlx lnqα .
l0
2.6
From 2.6, we have
α,β
Fχ,q t, x 2qβ t
∞
−1l qβl χlexlqα t α,β
Gn,χ,q x
n0
l0
α,β
∞
α,β
α,β
tn
.
n!
2.7
α,β
Observe that Fχ,q t Fχ,q t, 0. Hence we have Gn,χ,q Gn,χ,q 0. If q → 1 into 2.7, then
we easily obtain Fχ t, x.
α,β,r
First, we define the multiple generalized q-Genocchi numbers Gn,χ,q with weight α
and weak weight β:
α,β,r
Fχ,q t
2rqβ tr
−1
k1 ,...,kr 0
tr
∞
i1
ki β
q
r
i1
ki
r
χki e
r
i1
ki q α t
i1
· · · χx1 · · · χxr ex1 ···xr qα t dμ−qβ x1 · · · dμ−qβ xr X
X
r-times
r
∞
α,β,r t
Gn,χ,q
n0
n
n!
.
2.8
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5
Then we have
∞ tn
· · · χx1 · · · χxr x1 · · · xr nqα dμ−qβ x1 · · · dμ−qβ xr n!
n0 X X
2.9
r-times
∞
α,β,r t
Gn,χ,q
n−r
n!
n0
r−1
α,β,r t
Gn,χ,q
n−r
n!
n0
∞ Gα,β,r n
nr,χ,q t
,
nr
r r! n!
n0
where nr
r n r!/n!r!.
By comparing the coefficients on the both sides of 2.9, we obtain the following
theorem.
Theorem 2.1. Let q ∈ Cp with |1 − q|p < 1 and n ∈ Z . Then one has
α,β,r
α,β,r
α,β,r
G0,χ,q G1,χ,q · · · Gr−1,χ,q 0,
α,β,r
Gnr,χ,q
nr
r r!
· · · χx1 · · · χxr x1 · · · xr nqα dμ−qβ x1 · · · dμ−qβ xr X X
r-times
2rqβ
1 − qα
2rqβ
n
a1
r
r
r
f−1 n −1l i1 ai qαlβ i1 ai
n
χai r
l
1 qfαlβ
,...,a 0 l0
i1
r
n
f−1 r
r
r
r
mr−1
a
m
β
a
fm
×
χai ai fm .
−1 i1 i q i1 i
m
α
m0 a ,...,a 0
i1
i1
∞
1
q
r
2.10
α,β,r
From now on, we define the multiple generalized q-Genocchi polynomials Gn,χ,q x
with weight α and weak weight β.
α,β,r
Fχ,q t, x
2rqβ tr
tr
∞
−1
k1 ,...,kr 0
i1
ki β
q
r
i1
ki
r
χki e
r
i1
ki xqα t
i1
· · · χ y1 · · · χ yr exy1 ···yr qα t dμ−qβ y1 · · · dμ−qβ yr
X X
r-times
r
∞
α,β,r
Gn,χ,q x
n0
tn
.
n!
2.11
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Then we have
∞ n
tn
· · · χ y1 · · · χ yr x y1 · · · yr qα dμ−qβ y1 · · · dμ−qβ yr
n!
n0 X X
2.12
r-times
∞
α,β,r
Gn,χ,q x
n0
t
n−r
n!
r−1
α,β,r
Gn,χ,q x
n0
t
n−r
n!
∞ Gα,β,r x n
t
nr,χ,q
,
nr
r r! n!
n0
where nr
r n r!/n!r!.
By comparing the coefficients on the both sides of 2.12, we have the following
theorem.
Theorem 2.2. Let q ∈ Cp with |1 − q|p < 1 and n ∈ Z . Then one has
α,β,r
α,β,r
α,β,r
G0,χ,q x G1,χ,q x · · · Gr−1,χ,q x 0,
α,β,r
Gnr,χ,q x
nr
r r!
n
· · · χ y1 · · · χ yr x y1 · · · yr qα dμ−qβ y1 · · · dμ−qβ yr
X X
r-times
2rqβ
1 − qα
2rqβ
n
a1
2.13
r
f−1 r
r
mr−1
−1 i1 ai m qβ i1 ai fm
m
m0 a ,...,a 0
∞
1
×
r
r
f−1 r
n −1l i1 ai qαlxαlβ i1 ai
n
χai r
l
1 qfαlβ
,...,a 0 l0
i1
r
r
χai r
i1
n
ai fm x
i1
.
qα
In 2.11, we simply identify that
α,β,r
lim Fχ,q t, x
q→1
∞
2t
r r
−1
r
k1 ,...,kr 0
⎛
⎝
2t
f−1
i1 ki
r
χki e
r
i1
ki xt
i1
a
at
a0 −1 χae
1 eft
⎞r
2.14
⎠ etx Fχr t, x.
So far, we have studied the generating functions of the multiple generalized qα,β,r
α,β,r
Genocchi numbers Gn,χ,q and polynomials Gn,χ,q x with weight α and weak weight β.
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7
3. Modified Multiple Generalized q-Genocchi Polynomials with
Weight α and Weak Weight β
In this section, we will investigate about modified multiple generalized q-Genocchi numbers
and polynomials with weight α and weak weight β. Also, we will find their relations in
multiple generalized q-Genocchi numbers and polynomials with weight α and weak weight
β.
α,β,r
α,β,r
Firstly, we modify generating functions of Gn,χ,q and Gn,χ,q x. We access some
relations connected to these numbers and polynomials with weight α and weak weight β.
For this reason, we assign generating function of modified multiple generalized q-Genocchi
α,β,r
numbers and polynomials with weight α and weak weight β which are implied by Gn,χ,q
α,β,r
and Gn,χ,q x. We give relations between these numbers and polynomials with weight α and
weak weight β.
We modify 2.11 as follows:
α,β,r
Fχ,q
α,β,r −αx
t, x Fχ,q
q
t, x ,
3.1
α,β,r
where Fχ,q t, x is defined in 2.11.
From the above we know that
α,β,r
Fχ,q
t, x ∞
α,β,r
q−nrαx Gn,χ,q x
n0
tn
.
n!
3.2
t,
3.3
After some elementary calculations, we attain
α,β,r
Fχ,q
t, x q−αrx eq
−αx
xqα t
α,β,r
Fχ,q
α,β,r
where Fχ,q t is defined in 2.8.
From the above, we can assign the modified multiple generalized q-Genocchi
α,β,r
polynomials εn,χ,q x with weight α and weak weight β as follows:
α,β,r
Fχ,q
t, x ∞
tn
α,β,r
εn,χ,q x .
n!
n0
3.4
Then we have
α,β,r
α,β,r
εn,χ,q x q−nrαx Gn,χ,q x.
3.5
Theorem 3.1. For r ∈ N and n ∈ Z , one has
α,β,r
εn,χ,q x q−nrαx
n n
i0
i
α,β,r
qαix xn−i
qα Gi,χ,q .
3.6
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International Journal of Mathematics and Mathematical Sciences
Corollary 3.2. For r ∈ N and n ∈ Z , by using 3.7, one easily obtains
α,β,r
εn,χ,q x q−nrαx
n−j n ∞ m0 j0 l0
n
j, l, n − j − l
n−j m−1
α,β,r
−1l qα{jlxm} Gj,χ,q .
m
3.7
Secandly, by using generating function of the multiple generalized q-Genocchi
polynomials with weight α and weak weight β, which is defined by 2.11, we obtain the
following identities.
By using 2.13, we find that
α,β,r
Gnr,χ,q x
nr
r r!
2rqβ
f−1 r
mr−1
−1 i1 ai m
m
m0 a ,...,a 0
∞
1
×q
β
r
i1
r
ai fm
r
n
r
χai ai fm x
i1
i1
2rqβ
×
f−1
l n a1 ,...,ar 0 l0 a0
r
n
−1a
a, l − a, n − l
χai 1 − qα
i1
qα
r
l xn−l
qα
1 qf{αan−lβ}
i1
ai {αan−lβ}
q
3.8
r
i1
ai
r .
Thus we have the following theorem.
Theorem 3.3. Let q ∈ Cp with |1 − q|p < 1 and r ∈ N. Then one has
α,β,r
Gnr,χ,q x
nr
r r!
2rqβ
×
f−1
l n a1 ,...,ar 0 l0 a0
r
i1
r
r
n
−1a i1 ai q{αan−lβ} i1 ai
a, l − a, n − l
χai 1 − qα
l xn−l
qα
1 qf{αan−lβ}
3.9
r .
By using 2.13, we have
α,β,r
Fχ,q
f−1
l
n ∞ r
n −1 qαlx −1 i1 ai
n
l
1 − q a1 ,...,ar 0
n0 l0
r
∞
m r − 1 fαlβ m tn
αlβ ri1 ai .
×q
χai −q
m
n!
m0
i1
t, x 2rqβ tr
3.10
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9
Thus we have
∞
f−1
∞
n r
−n n
r r
2qβ t
−1 i1 ai
−1l qαlx 1 − qα
l
n! n0
a1 ,...,ar 0
l0
t
α,β,r
Gn,χ,q x
n0
n
× qαlβ
r
i1
ai r
χai 1 qfαlβ
−r tn
n!
i1
3.11
.
By comparing the coefficients of both sides of n r!/tnr in the above, we arrive at the
following theorem.
Theorem 3.4. Let q ∈ Cp with |1 − q|p < 1, r ∈ N. Then one has
α,β,r
Gnr,χ,q x
nr
r r!
2rqβ
n n
l
l0
× qαlβ
r
i1
f−1
r
−n −1 i1 ai
−1l qαlx 1 − qα
ai r
χai a1 ,...,ar 0
1 qfαlβ
−r
3.12
.
i1
From 2.12, we easily know that
∞
∞
∞
r
r
tn nr
α,β,r
Gn,χ,q x r!
2rqβ
−1 i1 ki qβ i1 ki
r
n! n0
n0
k ,...,k 0
1
r
× x ki
i1
r
r
χki i1
3.13
n
q
tnr
.
n r!
α
From the above, we get the following theorem.
Theorem 3.5. Let r ∈ N, k ∈ Z . Then one has
α,β,r
α,β,r
α,β,r
G0,χ,q x G1,χ,q x · · · Gr−1,χ,q x 0,
α,β,r
Glr,χ,q x
2rqβ
l
r
∞
r
r
r
lr
k
β
k
χki x ki .
r!
−1 i1 i q i1 i
r
α
i1
i1
k ,...,k 0
1
r
q
3.14
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From 2.13, we have
∞
∞
tn tn
α,β,r
α,β,s
Gn,χ,q x
Gn,χ,q x
n! n0
n!
n0
2rs
trs
qβ
×
r
∞ mr−1
f−1
m
a1 ,...,ar 0 m0
χai e
×q
s
i1 bi fk
f−1
r
i1 ai fmxqα t
s
r
i1
ai m β
q
χbi e
r
i1
r
i1
ai fm
∞ ks−1
k
b1 ,...,bs 0 k0
i1
β
−1
bi fkxqα t
−1
s
3.15
i1 bi k
.
i1
By using Cauchy product in 3.15, we obtain
∞ n n
n0 j0
j
α,β,r
α,β,s
Gj,χ,q xGn−j,χ,q x
2rs
trs
qβ
tn
n!
f−1 j r−1
n−j s−1
j
n−j
n0 j0 a ,...,a 0 b ,...,b 0
n
∞ f−1
1
r
1
s
s
r
r
s
r
s
i1 ai i1 bi n β i1 ai i1 bi fn
q
χai χbi × −1
i1
× e
r
i1
ai fjxqα t e
s
i1
bi fn−jxqα t
3.16
i1
.
From 3.16, we have
⎞
⎛
m tm
m
α,β,r
α,β,s
⎝
Gj,χ,q xGm−j,χ,q x⎠
j
m!
m0
j0
∞
∞
trs
2rs
qβ
m0
× −1
r
i1
n0 j0 a1 ,...,ar 0 b1
f−1 j r−1
n−j s−1
j
n−j
,...,b 0
s
s
n
∞ ai i1
f−1
bi n β
q
r
i1
ai s
r
s
i1 bi fn
χai χbi i1
3.17
i1
⎛
⎞m
r
s
tm
×⎝
.
ai fj x
bi fn − j x ⎠
m!
α
α
i1
i1
q
q
By comparing the coefficients of both sides of tmrs /m r s! in 3.17, we have the
following theorem.
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11
Theorem 3.6. Let r ∈ Nand s ∈ Z . Then one has
lrs lrs j
j0
α,β,r
α,β,s
Gj,χ,q xGlrs−j,χ,q x
lrs r s!
l
2rs
qβ
n
∞ f−1
n0 j0 a1 ,...,ar 0 b1
× −1
r
i1 ai s
f−1 j r−1
n−j s−1
j
n−j
,...,b 0
i1 bi n β
q
s
r
i1
ai s
r
i1 bi fn
χai χbi s
i1
3.18
i1
⎛
⎞l
r
s
×⎝
ai fj x
bi f n − j x ⎠ .
i1
i1
qα
qα
Corollary 3.7. In 3.18 setting s 1, one has
lr1 lr1 α,β,r
α,β,1
Gj,χ,q xGlr1−j,χ,q x
lr1 r 1!
l
j
j0
2r1
qβ
∞ n
f−1
f−1 j r−1
j
n0 j0 a1 ,...,ar 0 b1 0
−1
r
i1
ai b1 n β
q
r
i1
ai b1 fn
3.19
⎞l
⎛
r
r
χb1 χai ⎝
ai fj x
b1 f n − j x qα ⎠ .
×
i1
i1
qα
By using 2.13 we have the following theorem.
Theorem 3.8. Distribution theorem is as follows:
α,β,r
Gnr,χ,q
n
f−1
r
f qα r
r
α,β,r a1 · · · ar
ai
i1 ai β
i1
,
r
q
χai Gnr,qf
−1
f
f −qβ a1 ,...,ar 0
i1
n
r
f−1
f qα r
r
α,β,r
α,β,r x a1 · · · ar
a
β
a
i
Gnr,χ,q x r
χai Gnr,qf
.
−1 i1 q i1 i
f
f −qβ a1 ,...,ar 0
i1
3.20
4. Interpolation Function of Multiple Generalized q-Genocchi
Polynomials with Weight α and Weak Weight β
In this section, we see interpolation function of multiple generalized q-Genocchi polynomials
with weak weight α and find some relations.
12
International Journal of Mathematics and Mathematical Sciences
α,β,r
Let us define interpolation function of the Gkr,q x as follows.
Definition 4.1. Let q, s ∈ C with |q| < 1 and 0 < x ≤ 1. Then one defines
α,β,r
ζχ,q s, x
2rqβ
∞
−1
r
i1
ki β
k1 ,...,kr 0
q
r
x
r
i1 χki s
r
i1 ki qα
i1
ki
4.1
.
α,β,r
We call ζq
s, x the multiple generalized Hurwitz type q-zeta funtion.
In 4.1, setting r 1, we have
α,β,1
ζχ,q
s, x 2qβ
∞
−1l qβl χl
x l0
α,β
ζχ,q s, x.
4.2
ki r
i1 χki .
s
ri1 ki
4.3
lsqα
Remark 4.2. It holds that
α,β,r
lim ζχ,q s, x
q→1
r
∞
2
r
−1
i1
x
k1 ,...,kr 0
Substituting s −n, n ∈ Z into 4.1, then we have,
α,β,r
ζχ,q −n, x
2rqβ
∞
−1
r
i1
ki β
q
r
i1 ki
r
n
r
χki x ki .
k1 ,...,kr 0
i1
i1
4.4
qα
Setting 3.14 into the above, we easily get the following theorem.
Theorem 4.3. Let r ∈ N, n ∈ Z . Then one has
α,β,r
ζχ,q −n, x
α,β,r
Gnr,χ,q x
nr
r r!
.
4.5
References
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