Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2008, Article ID 304539, 13 pages
doi:10.1155/2008/304539
Research Article
Multivariate p-Adic Fermionic q-Integral on Zp
and Related Multiple Zeta-Type Functions
Min-Soo Kim,1 Taekyun Kim,2 and Jin-Woo Son1
1
2
Department of Mathematics, Kyungnam University, Masan 631-701, South Korea
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
Correspondence should be addressed to Min-Soo Kim, mskim@kyungnam.ac.kr
Received 14 April 2008; Accepted 27 May 2008
Recommended by Ferhan Atici
In 2008, Jang et al. constructed generating functions of the multiple twisted Carlitz’s type qBernoulli polynomials and obtained the distribution relation for them. They also raised the
following problem: “are there analytic multiple twisted Carlitz’s type q-zeta functions which interpolate
multiple twisted Carlitz’s type q-Euler (Bernoulli) polynomials?” The aim of this paper is to give a partial
answer to this problem. Furthermore we derive some interesting identities related to twisted qextension of Euler polynomials and multiple twisted Carlitz’s type q-Euler polynomials.
Copyright q 2008 Min-Soo Kim 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.
1. Introduction, definitions, and notations
Let p be an odd prime. Zp , Qp , and Cp will always denote, respectively, the ring of p-adic
integers, the field of p-adic numbers, and the completion of the algebraic closure of Qp . Let
vp : Cp →Q ∪ {∞} Q is the field of rational numbers denote the p-adic valuation of Cp
normalized so that vp p 1. The absolute value on Cp will be denoted as |·|p , and |x|p p−vp x
for x ∈ Cp . We let Z×p {x ∈ Zp | 1/x ∈ Zp }. A p-adic integer in Z×p is sometimes called a p-adic
unit. For each integer N ≥ 0, CpN will denote the multiplicative group of the primitive pN th
roots of unity in C×p Cp \ {0}. Set
N
Tp ω ∈ Cp | ωp 1 for some N ≥ 0 Cp N .
1.1
N≥0
The dual of Zp , in the sense of p-adic Pontrjagin duality, is Tp Cp∞ , the direct limit under
inclusion of cyclic groups CpN of order pN N ≥ 0, with the discrete topology.
2
Abstract and Applied Analysis
When one talks of q-extension, q is variously considered as an indeterminate, a complex
number q ∈ C, or a p-adic number q ∈ Cp . If q ∈ Cp , then we normally assume |1−q|p < p−1/p−1 ,
so that qx expx log q for |x|p ≤ 1. If q ∈ C, then we assume that |q| < 1.
Let
Z
,
Zp lim N
Z×p a pZp ,
←
p Z
0<a<p
N
1.2
N
N
,
a p Zp x ∈ Zp | x ≡ a mod p
where a ∈ Z lies in 0 ≤ a < pN .
We use the following notation:
xq 1 − qx
.
1−q
1.3
Hence
limxq x
1.4
q→1
for any x with |x|p ≤ 1 in the present p-adic case. The distribution μq a pN Zp is given as
μq a pN Zp qa
pN
1.5
q
cf. 1–9. For the ordinary p-adic distribution μ0 defined by
1
μ0 a pN Zp N ,
p
1.6
lim μq μ0 .
1.7
we see
q→1
We say that f is a uniformly differentiable function at a point a ∈ Zp , we write f ∈ UDZp , Cp if the difference quotient
Ff x, y fx − fy
x−y
1.8
has a limit f a as x, y→a, a. Also we use the following notation:
x−q cf.1–5.
1 − −qx
,
1q
1.9
Min-Soo Kim et al.
3
In 1–3, Kim gave a detailed proof of fermionic p-adic q-measures on Zp . He treated
some interesting formulae-related q-extension of Euler numbers and polynomials; and he
defined fermionic p-adic q-measures on Zp as follows:
−qa
μ−q a pN Zp N .
p −q
1.10
By using the fermionic p-adic q-measures, he defined the fermionic p-adic q-integral on Zp as
follows:
I−q f Zp
fxdμ−q x lim
N→∞
pN −1
1
pN
fx−qx
1.11
−q x0
for f ∈ UDZp , Cp cf. 1–3. Observe that
pN −1
I−1 f limI−q f q→1
Zp
fx−1x .
fxdμ−1 x lim
N→∞
1.12
x0
From 1.12, we obtain
I−1 f1 I−1 f 2f0,
1.13
where f1 x fx 1. By substituting fx etx into 1.13, classical Euler numbers are
defined by means of the following generating function:
Zp
etx dμ−1 x 2
t
e 1
∞
En
n0
tn
.
n!
1.14
These numbers are interpolated by the Euler zeta function which is defined as follows:
ζE s ∞
−1n
,
ns
n1
s ∈ C,
1.15
cf. 1–9. From 1.12, we also obtain
qI−q f1 I−q f 2q f0,
1.16
where f1 x fx 1. By substituting fx etx into 1.13, q-Euler numbers are defined by
means of the following generating function:
Zp
etx dμ−q x 2q
qet
1
∞
En,q
n0
tn
.
n!
1.17
4
Abstract and Applied Analysis
These numbers are interpolated by the Euler q-zeta function which is defined as follows:
ζq,E s 2q
∞
−1n qn
,
ns
n1
s ∈ C,
1.18
cf. 4.
In 6, Ozden and Simsek defined generating function of q-Euler numbers by
2
q1
Zp
etx dμ−q x 2
,
qet 1
1.19
which are different from 1.17. But we observe that all these generating functions were
obtained by the same fermionic p-adic q-measures on Zp and the fermionic p-adic q-integrals
on Zp .
In this paper, we define a multiple twisted Carlitz’s type q-zeta functions, which
interpolated multiple twisted Carlitz’s type q-Euler polynomials at negative integers. This
result gave us a partial answer of the problem proposed by Jang et al. 10, which is given by:
“Are there analytic multiple twisted Carlitz’s type q-zeta functions which interpolate multiple twisted
Carlitz’s type q-Euler (Bernoulli) polynomials?”
2. Preliminaries
In 10, Jang and Ryoo defined q-extension of Euler numbers and polynomials of higher order
and studied multivariate q-Euler zeta functions. They also derived sums of products of q-Euler
numbers and polynomials by using ferminonic p-adic q-integral.
In 5, 7, Ozden et al. defined multivariate Barnes-type Hurwitz q-Euler zeta functions
and l-functions. They also gave relation between multivariate Barnes-type Hurwitz q-Euler
zeta functions and multivariate q-Euler l-functions.
In this section, we consider twisted q-extension of Euler numbers and polynomials of
higher order and study multivariate twisted Barnes-type Hurwitz q-Euler zeta functions and
l-functions.
Let UDZhp , Cp denote the space of all uniformly or strictly differentiable Cp -valued
functions on Zhp Zp × · · · × Zp . For f ∈ UDZhp , Cp , the p-adic q-integral on Zhp is defined by
h-times
h
I−q f Zp
···
Zp
f x1 , . . . , xh dμ−q x1 · · · dμ−q xh
h-times
lim
N→∞
1
2.1
pN −1
h
pN −q x1 0
pN −1
f x1 , . . . , xh −qx1 ···xh
···
xh 0
cf. 3. If q→1, then
h
h
pN −1
I−1 f limI−q f lim
q→1
N→∞
pN −1
f x1 , . . . , xh −1x1 ···xh .
···
x1 0
xh 0
2.2
Min-Soo Kim et al.
5
For a fixed positive integer d with d, p 1, we set
Xp lim
←
−
N
Z
.
dpN Z
2.3
For f ∈ UDZhp , Cp ,
h
I−1 f f x1 , . . . , xh dμ−1 x1 · · · dμ−1 xh ,
···
Xp
2.4
Xp
h-times
cf. 2.
We set fx1 , . . . , xh ωx1 ···xh exx1 ···xh t in 2.2 and 2.4. Then we have
Zp
···
Zp
ωx1 ···xh exx1 ···xh t dμ−1 x1 · · · dμ−1 xh 2
ωet 1
2
···
ωet 1
∞
h
En,ω x
n0
tn
,
n!
h-times
h-times
2.5
h
where En,ω x are the twisted Euler polynomials of order h. From 2.5, we note that
Zp
···
Zp
n
h
ωx1 ···xh x x1 · · · xh dμ−1 x1 · · · dμ−1 xh En,ω x.
2.6
h-times
We give an application of the multivariate q-deformed p-adic integral on Zhp in the
fermionic sense related to 3. Let
Zhp
Zp
···
Zp
2.7
.
h-times
By substituting
f x1 , . . . , xh ωx1 ···xh exx1 ···xh t
2.8
into 2.1, we define twisted q-extension of Euler numbers of higher order by means of the
following generating function:
Zhp
ωx1 ···xh ex1 ···xh t dμ−q x1 · · · dμ−q xh 2 q
·
·
·
t
t
ωqe 1
ωqe 1
2q
h-times
∞
h
En,q,ω
n0
tn
.
n!
2.9
6
Abstract and Applied Analysis
Then we have
Zhp
n
h
ωx1 ···xh x1 · · · xh dμ−q x1 · · · dμ−q xh En,q,ω .
2.10
From 2.9, we obtain
2hq ext
ωx1 ···xh exx1 ···xh t dμ−q x1 · · · dμ−q xh ωqet 1 · · · ωqet 1
Zhp
∞
h
En,q,ω x
n0
h-times
tn
,
n!
2.11
h
where En,q,ω x is called twisted q-extension of Euler polynomials of higher order cf. 11. We
h
h
h
h
note that if ω 1, then En,q,ω x En,q x and En,q,ω En,q cf. 6. We also note that
n
h
Ek,q,ω xn−k .
k
k0
n
h
En,q,ω x
2.12
h
The twisted q-extension of Euler polynomials of higher order, En,q,ω x, is defined by
means of the following generating function:
2q
···
ext
1
ωqet 1
2q
h
Gq,ω x, t ωqet
h-times
2hq etx
2hq
∞
∞
−ωl1 ql1 el1 t · · ·
l1 0
∞
∞
−ωlh qlh elh t
lh 0
2.13
−ωl1 ···lh ql1 ···lh el1 ···lh xt
l1 ,...,lh 0
h
En,q,ω x
n0
tn
,
n!
where |t logωq| < π. From these generating functions of twisted q-extension of Euler
polynomials of higher order, we construct twisted multiple q-Euler zeta functions as follows.
For s ∈ C and x ∈ R with 0 < x ≤ 1, we define
h
ζq,ω,E s, x 2hq
−ωl1 ···lh ql1 ···lh
s .
l1 ,...,lh 0 l1 · · · lh x
∞
2.14
By the mth differentiation on both sides of 2.13 at t 0, we obtain the following
h
Em,q,ω x for m 0, 1, . . . .
d
dt
m
h
Gq,ω x, t
t0
2hq
∞
−ωl1 ···lh ql1 ···lh x l1 · · · lh m
l1 ,...,lh 0
2.15
Min-Soo Kim et al.
7
From 2.14 and 2.15, we arrive at the following
h
h
ζq,ω,E −m, x Em,q,ω x,
m 0, 1, . . . .
2.16
.
2.17
We set
Xph
···
Xp
Xp
h-times
Let χ be Dirichlet’s character with odd conductor d. We define twisted q-extension of
generalized Euler polynomials of higher order by means of the following generating function
cf. 11:
Xph
χ x1 · · · xh ωx1 ···xh exx1 ···xh t dμ−q x1 · · · dμ−q xh ∞
h
En,q,ω,χ x
n0
tn
.
n!
2.18
Note that
∞
h
En,q,ω,χ x
n0
tn
n!
χ x1 · · · xh ωx1 ···xh ex1 ···xh t dμ−q x1 · · · dμ−q xh
···
ext
Xp
Xp
h-times
ext
1
N
d−1 p −1
1
lim
dh−q N→∞
pN
···
−q a1 0 x1 0
N
d−1 p −1
χ a1 dx1 · · · ah dxh
ah 0 xh 0
d
× ωa1 dx1 ···ah dxh ea1 dx1 ···ah dxh t −qa1 dx1 ···ah dxh
ext
1
d−1
χ a1 · · · ah ωa1 ···ah −qa1 ···ah ea1 ···ah t
2.19
dh−q a1 ,...,ah 0
N
N
N
N
N
N
1 qd 1 ωdp qdp edp
1 qd 1 ωdp qdp edp
·
·
·
lim
N
N
N→∞ 1 qdp
N→∞ 1 qdp
1 ωd qd edt
1 ωd qd edt
× lim
h−times
ext
×
1
d−1
χ a1 · · · ah ωa1 ···ah −qa1 ···ah ea1 ···ah t
dh−q a1 ,...,ah 0
1 qd
1 qd
·
·
·
1 ωd qd edt
1 ωd qd edt
h-times
since
N
lim qp 1 for |1 − q|p < 1.
N→∞
2.20
8
Abstract and Applied Analysis
This allows us to rewrite 2.18 as
∞
h
En,q,ω,χ x
n0
ext
×
tn
n!
χ a1 · · · ah ωa1 ···ah −qa1 ···ah ea1 ···ah t
d−1
1
dh−q a1 ,...,ah 0
1 qd
1 qd
·
·
·
1 ωd qd edt
1 ωd qd edt
h-times
2hq ext
×
∞
d−1
χ a1 · · · ah ωa1 ···ah −qa1 ···ah ea1 ···ah t
a1 ,...,ah 0
− ωd qd edt
x1
x1 0
···
∞
− ωd qd edt
xh 0
2.21
xh
h-times
2hq ext
∞
χ a1 dx1 · · · ah dxh
d−1
x1 ,...,xh 0 a1 ,...,ah 0
× ωa1 dx1 ···ah dxh −qa1 dx1 ···ah dxh ea1 dx1 ···ah dxh t
2hq
∞
−1l1 ···lh χ l1 · · · lh ωl1 ···lh ql1 ···lh exl1 ···lh t .
l1 ,...,lh 0
By applying the mth derivative operator d/dtm |t0 in the above equation, we have
∞
h
Em,q,ω,χ x 2hq
h
m
χ l1 · · · l h
−1li ωli qli x l1 · · · lh
l1 ,...,lh 0
2.22
i1
for m 0, 1, . . . .
From these generating functions of twisted q-extension of generalized Euler polynomials
of higher order, we construct twisted multiple q-Euler l-functions as follows. For s ∈ C and
x ∈ R with 0 < x ≤ 1, we define
h
lq,ω,E s, x, χ
2hq
h
li li li
χ l1 · · · l h
i1 −1 ω q
.
s
l1 · · · l h x
l1 ,...,lh 0
∞
2.23
From 2.22 and 2.23, we arrive at the following
h
h
lq,ω,E −m, x, χ Em,q,ω,χ x,
m 0, 1, . . . .
2.24
Min-Soo Kim et al.
9
Let s ∈ C and ai , F ∈ Z with F is an odd integer and 0 < ai < F, where i 1, . . . , h. Then
twisted partial multiple q-Euler ζ-functions are as follows:
h Hq,ω,E s, a1 , . . . , ah , x | F 2hq
∞
l1 ,...,lh 0
li ≡ai mod F, i1,...,h
−1l1 ···lh ωl1 ···lh ql1 ···lh
.
s
l1 · · · l h x
2.25
For i 1, . . . , h, substituting li ai ni F with F is odd into 2.25, we have
h Hq,ω,E s, a1 , . . . , ah , x | F
2hq
−1a1 n1 F···ah nh F ωa1 n1 F···ah nh F qa1 n1 F···ah nh F
s
a1 n1 F · · · ah nh F x
n1 ,...,nh 0
∞
n ···nh F n1 ···nh
∞
2hq −ωqa1 ···ah h
−1n1 ···nh ωF 1
q
2qF
s
s
h
F
2 F
n1 ,...,nh 0 n1 · · · nh a1 · · · ah x /F
2.26
q
2hq −ωqa1 ···ah h
a1 · · · ah x
.
ζqF ,ωF ,E s,
Fs
F
2hF
q
Then we obtain
h Hq,ω,E s, a1 , . . . , ah , x
2hq −ωqa1 ···ah h
a1 · · · ah x
.
|F ζqF ,ωF ,E s,
Fs
F
2hF
2.27
q
By using 2.12 and 2.27 and by substituting s −m, m 0, 1, . . . , we get
h
2q
m
h Hq,ω,E − m, a1 , . . . , ah , x | F −ωqa1 ···ah a1 · · · ah x
h
2qF
k
m
F
m
h
×
Ek,qF ,ωF .
k
a
·
·
·
a
x
1
h
k0
2.28
Therefore, we modify twisted partial multiple q-Euler zeta functions as follows:
h
2q
−s
h Hq,ω,E s, a1 , . . . , ah , x | F −ωqa1 ···ah a1 · · · ah x
h
2qF
k
∞
F
−s
h
×
Ek,qF ,ωF .
k
a
·
·
·
a
x
1
h
k0
2.29
Let χ be a Dirichlet character with conductors d and d | F. From 2.23 and 2.27, we have
h
lq,ω,E s, x, χ 2hq
2hqF
F −s
F−1
−ωqa1 ···ah
a1 ,...,ah 0
h
a1 · · · ah x
× χ a1 · · · ah ζqF ,ωF ,E s,
F
F−1
h χ a1 · · · ah Hq,ω,E s, x, a1 , . . . , ah , x | F .
a1 ,...,ah 0
2.30
10
Abstract and Applied Analysis
3. The multiple twisted Carlitz’s type q-Euler polynomials and q-zeta functions
Let us consider the multiple twisted Carlitz’s type q-Euler polynomials as follows:
z,h
En,q,ω x Zp
···
n
x1 · · · xh x q ωx1 ···xh qx1 z−1···xh z−h dμq x1 · · · dμq xh
Zp
3.1
h-times
cf. 1, 3. These can be written as
z,h
En,q,ω x
1
1
n
qix −1i
···
.
1 ωqzi
1 − qn i0 i
1 ωqzi−h1
2hq
n
3.2
We may now mention the following formulae which are easy to prove:
z,h
z,h
z−1,h−1
ωqz En,q,ω x 1 En,q,ω x 2q En,q,ω
x.
3.3
From 3.2, we can derive generating function for the multiple twisted Carlitz’s type q-Euler
polynomials as follows:
∞
z,h
En,q,ω x
n0
tn
n!
1
tn
1
n
·
·
·
qix −1i
n
i
1 ωqzi
1 ωqzi−h1 n!
n0 1 − q i0
2hq
∞
n
∞
∞
lh tn
n
i
ix
zi l1
−1
·
·
·
q
−
ωq
− ωqzi−h1
n
i
n!
n0 1 − q i0
l 0
l 0
2hq
∞
n
1
h
∞
∞
n
2hq
n
l1 ···lh
qxl1 ···lh i −1i
−1
n
i
1
−
q
n0
i0
l ,...,l 0
1
h
× ωl1 ···lh ql1 zl2 z−1···lh z−h1
2hq
×
∞
t
n!
−1l1 ···lh ωl1 ···lh ql1 zl2 z−1···lh z−h1
l1 ,...,lh 0
∞
x l1 · · · lh
n0
2hq
3.4
n
∞
n
nt
q n!
−ωl1 ···lh ql1 zl2 z−1···lh z−h1 exl1 ···lh q t .
l1 ,...,lh 0
Also, an obvious generating function for the multiple twisted Carlitz’s type q-Euler
polynomials is obtained, from 3.2, by
j
n
1
1
1
z,h
2hq et/1−q −1j qjx
···
En,q,ω x.
3.5
zj
zj−h1
1
−
q
1
ωq
1
ωq
j0
Min-Soo Kim et al.
11
From now on, we assume that q ∈ C with |q| < 1. From 3.2 and 3.4, we note that
z,h
Gq,ω x, t
2hq
∞
∞
−ωl1 ···lh ql1 zl2 z−1···lh z−h1 exl1 ···lh q t
3.6
l1 ,...,lh 0
z,h
En,q,ω x
n0
tn
,
n!
z,h
En,q,ω x
1
1
n
···
qix −1i
1 ωqzi
1 − qn i0 i
1 ωqzi−h1
2hq
2hq
n
∞
3.7
−ωl1 ···lh ql1 zl2 z−1···lh z−h1 x l1 · · · lh q .
n
l1 ,...,lh 0
Thus we can define the multiple twisted Carlitz’s type q-zeta functions as follows:
z,h
ζq,ω s, x 2hq
−ωl1 ···lh ql1 zl2 z−1···lh z−h1
∞
x l1 · · · lh
l1 ,...,lh 0
s
q
3.8
.
z,h
In 12, Proposition 3, Yamasaki showed that the series ζq,ω s, x converges absolutely
for Re z > h − 1, and it can be analytically continued to the whole complex plane C. Note that
if h 1, then
z,h
z
ζq,ω s, x −→ ζq,ω s, x 2q
∞
−ωl qlz
.
x lsq
l0
3.9
In 13, Wakayama and Yamasaki studied q-analogue of the Hurwitz zeta function
ζs, x ∞
1
s
n0 n x
3.10
defined by the q-series with two complex variable s, z ∈ C:
z
ζq s, x ∞
qnxz
s,
n0 x nq
3.11
Rez > 0,
and special values at nonpositive integers of the q-analogue of the Hurwitz zeta function.
Therefore, by the mth differentiation on both sides of 3.6 at t 0, we obtain the
following:
m
d
z,h
z,h
Em,q,ω x Gq,ω x, t
dt
t0
3.12
∞
2hq
for m 0, 1, . . . .
−ωl1 ···lh ql1 zl2 z−1···lh z−h1 x l1 · · · lh
l1 ,...,lh 0
m
q
12
Abstract and Applied Analysis
From 3.7, 3.8, and 3.12, we have 3.13 which shows that the multiple twisted
Carlitz’s type q-zeta functions interpolate the multiple twisted Carlitz’s type q-Euler numbers
and polynomials. For m 0, 1, . . . , we have
z,h
z,h
ζq,ω −m, x Em,q,ω x,
3.13
where x ∈ R and 0 < x ≤ 1.
Thus, we derive the analytic multiple twisted Carlitz’s type q-zeta functions which
interpolate multiple twisted Carlitz’s type q-Euler polynomials. This gives a part of the answer
to the question proposed in 10.
4. Remarks
For nonnegative integers m and n, we define the q-binomial coefficient
m
n q
by
q; qm
m
,
n
q; qn q; qm−n
4.1
q
where a; qm m−1
k0
1 − aqk for m ≥ 1 and a; q0 1. For h ∈ N, it holds that
q
−l1 2l2 ···hlh q
−lh
l1 ,...,lh ≥0
l1 ···lh l
lh−1
h−1
4.2
q
cf. 12, Lamma 2.3. From 3.8, it is easy to see that
z,h
ζq,ω s, x 2hq
2hq
2hq
−ωl1 ···lh qz1l1 ···lh −l1 2l2 ···hlh ∞
∞
−ωl qz1l
l xsq
l0
∞
l0
lh−1
h−1
s
q
x l1 · · · lh
l0 l1 ,...,lh ≥0
l1 ···lh l
q−l1 2l2 ···hlh 4.3
l1 ,...,lh ≥0
l1 ···lh l
−ωl qz−h1l
.
l xsq
q
We set mq ! mq m − 1q · · · 1q for m ∈ N. The following identity has been studied in 12:
lh−1
h−1
q
h−1
1
l xq − qlj x − jq h − 1q ! j1
h−1
k0
k
qlh−1−k Pq,h
xl xkq ,
4.4
k
x, 0 ≤ k ≤ h − 1, is a function of x defined by
where Pq,h
k
Pq,h
x −1h−1−k
h − 1q ! 1≤m <···<m
1
qm1 ···mh−1−k x − m1
h−1−k ≤h−1
q
· · · x − mh−1−k
q
4.5
Min-Soo Kim et al.
13
h−1
for 0 ≤ k ≤ h − 2 and Pq,h
x 1/h − 1q !. By using 3.9, 4.3, and 4.5, we have
z,h
ζq,ω s, x 2hq
h−1
k0
k
Pq,h
x
∞
−ωl qz−kl
l0
l xs−k
q
2h−1
q
h−1
k0
z−k
k
Pq,h
xζq,ω s − k, x,
4.6
and so
z,h
ζq,ω −m, x 2h−1
q
h−1
k0
z−k
k
Pq,h
xζq,ω −m − k, x.
4.7
z,h
The values of ζq,ω −m, x at h 2, 3 are given explicitly as follows:
z,2
z−1
z
ζq,ω −m, x 1 q ζq,ω −m − 1, x − qx − 1q ζq,ω −m, x ,
z,3
z−2
ζq,ω −m, x 1 q ζq,ω −m − 2, x
z−1
− qx − 1q q2 x − 2q ζq,ω −m − 1, x
z
q3 x − 1q x − 2q ζq,ω −m, x .
4.8
Acknowledgment
This work is supported by Kyungnam University Foundation Grant no. 2007.
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