Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 826548, 8 pages
doi:10.1155/2010/826548
Research Article
On the Symmetric Properties of
the Multivariate p-Adic Invariant Integral on Zp
Associated with the Twisted Generalized Euler
Polynomials of Higher Order
Taekyun Kim,1 Byungje Lee,2 and Young-Hee Kim1
1
2
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Department of Wireless Communications Engineering, Kwangwoon University,
Seoul 139-701, Republic of Korea
Correspondence should be addressed to Young-Hee Kim, yhkim@kw.ac.kr
Received 6 November 2009; Revised 11 March 2010; Accepted 14 March 2010
Academic Editor: Ulrich Abel
Copyright q 2010 Taekyun 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.
We study the symmetric properties for the multivariate p-adic invariant integral on Zp related to
the twisted generalized Euler polynomials of higher order.
1. Introduction
Let p be a fixed prime number. Throughout this paper, the symbols Z, Zp , Qp , and Cp denote
the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers,
and the completion of algebraic closure of Qp , respectively. Let N be the set of natural
numbers. The normalized valuation in Cp is denoted by | · |p with |p|p 1/p. Let UDZp be the space of uniformly differentiable function on Zp . For f ∈ UDZp , the fermionic p-adic
invariant integral on Zp is defined as
I f Zp
fxdμx lim
N
−1
p
fxμ x pN Zp lim
fx−1x
N
−1
p
N →∞
N →∞
x0
1.1
x0
see 1–25. For n ∈ N, we note that
Zp
fx ndμx −1
n
Zp
fxdμx n−1
−1n−1−l fl
l0
see 5.
1.2
2
Journal of Inequalities and Applications
Let d be a fixed odd positive integer. For N ∈ N, we set
Z/dpN Z,
X Xd lim
←
−
N
X∗ X1 Zp ,
a dpZp ,
1.3
0<a<dp
a,p1
a dpN Zp x ∈ X | x ≡ a mod dpN ,
where a ∈ Z lies in 0 ≤ a < dpN see 1–13. It is well known that for f ∈ UDZp ,
fxdμx X
Zp
1.4
fxdμx.
n
For n ∈ N, let Cpn be the cyclic group of order pn . That is, Cpn {ξ | ξp 1}. The p-adic
locally constant space, Tp , is defined by Tp limn → ∞ Cpn n≥1 Cpn .
Let χ be Dirichlet’s character with conductor d ∈ N and let ξ ∈ Tp . Then the generalized
twisted Bernoulli polynomials Bn,χ,ξ x attached to χ are defined as
t
d−1
a at
a0 χaξ e
ext
d
dt
ξ e −1
∞
tn
Bn,χ,ξ x
n!
n0
see 10.
1.5
In 4, 7, 10–12, the generalized twisted Bernoulli polynomials of order k attached to χ
are also defined as follows:
d−1
d−1
∞
t a0 χaξa eat
t a0 χaξa eat
tn
k
xt
×
·
·
·
×
e
Bn,χ,ξ x .
d
dt
d
dt
n!
ξ e −1
ξ e −1
n0
1.6
k-times
Recently, the symmetry identities for the generalized twisted Bernoulli polynomials
and the generalized twisted Bernoulli polynomials of order k are studied in 4, 12.
In this paper, we study the symmetric properties of the multivariate p-adic invariant
integral on Zp . From these symmetric properties, we derive the symmetry identities for
the twisted generalized Euler polynomials of higher order. In 14, Kim gave the relation
between the power sum polynomials and the generalized higher-order Euler polynomials.
The main purpose of this paper is to give the symmetry identities for the twisted generalized
Euler polynomials of higher order using the symmetric properties of the multivariate p-adic
invariant integral on Zp .
Journal of Inequalities and Applications
3
2. Symmetry Identities for the Twisted Generalized
Euler Polynomials of Higher Order
Let χ be Dirichlet’s character with an odd conductor d ∈ N. That is, d ∈ N with d ≡ 1 mod 2.
For ξ ∈ Tp , the twisted generalized Euler polynomials attached to χ, En,χ,ξ x, are defined as
2
χ y ξy exyt dμ y d−1
a0
X
∞
−1a χaξa eat xt tn
e
E
x
n,χ,ξ
n!
ξd edt 1
n0
see 12.
2.1
In the special case x 0, En,χ,ξ En,χ,ξ 0 are called the nth twisted generalized Euler numbers
attached to χ.
From 2.1, we note that
m χ y ξy x y dμ y Em,χ,ξ x,
m ∈ N ∪ {0}.
2.2
X
For n ∈ N with n ≡ 1 mod 2, we have
χxξx exndt dμx X
Let Tk,χ,ξ n χxξx ext dμx 2
X
n
l0
nd−1
−1l χl ξl elt .
2.3
l0
−1l χlξl lk . Then we see that
χxξx exndt dμx ξnd
X
χxξx ext dμx
X
∞
tk
ndxt ndx
2 Tk,χ,ξ nd − 1 .
k!
e
ξ dμx
k0
X
2
X
ext χxξx dμx
2.4
k
Now we define the twisted generalized Euler polynomials En,χ,ξ x of order k attached
to χ as follows:
d−1
k
∞
2 a0 −1a χaξa eat
tn
k
e
En,χ,ξ x .
d
dt
n!
ξ e 1
n0
xt
k
k
2.5
In the special case x 0, En,χ,ξ En,χ,ξ 0 are called the nth twisted generalized Euler numbers
of order k.
4
Journal of Inequalities and Applications
Let w1 , w2 , d ∈ N with w1 ≡ 1, w2 ≡ 1, and d ≡ 1 mod 2. Then we set
m
Jχ,ξ w1 , w2
Xm
| x m
i1 χxi m
ξ
i1
X
m
×
Xm
xi w1 e
m
xi w2 xw1 t
i1
dμx1 · · · dμxm ξdw1 w2 x edw1 w2 xt dμx
χxi ξ
m
i1
xi w2 e
m
i1
xi w1 yw2 t
2.6
dμx1 · · · dμxm ,
i1
where
· · · fx1 , . . . , xm dμx1 · · · dμxm .
X X
Xm
fx1 , . . . , xm dμx1 · · · dμxm 2.7
m-times
From 2.6, we note that
m
Jχ,ξ w1 , w2
m
| x Xm
×
χxi ξ
m
xi w1 e
i1
m
xi w1 t
i1
dμx1 · · · dμxm ew1 w2 xt
i1
χxm ξw2 xm ew2 xm t dμxm ξdw1 w2 x edw1 w2 xt dμx
X
X
m−1
×
X m−1
χxi ξ
m−1
i1
2.8
ew1 w2 yt
xi w2 e
m−1
i1
xi w2 t
dμx1 · · · dμxm−1 .
i1
From 2.4, we can easily derive the following equation:
dw
∞
1 −1
χxξx ext dμx
tk
l
l lt
X
.
χlξ
e
T
−
1
−1
dw
k,χ,ξ
1
k!
ξdw1 x edw1 xt dμx
l0
k0
X
2.9
It is not difficult to show that
e
m
w1 w2 xt
Xm
χxi ξ
m
i1 xi w1 e
m
i1 xi w1 t
dμx1 · · · dμxm i1
d−1
m
∞
w1k tk
2 a0 −1a χaξaw1 eaw1 t
m
w1 w2 xt
.
e
E
x
w
w
2
k,χ,ξ 1
k!
ξdw1 edw1 t 1
k0
2.10
Journal of Inequalities and Applications
5
By 2.8, 2.9, and 2.10, we see that
m
Jχ,ξ w1 , w2
| x ∞
l0
×
w l tl
m
El,χ,ξw1 w2 x 1
l!
∞
i0
∞
n0
w i ti
m−1 Ei,χ,ξw2 w1 y 2
i!
∞
k0
w k tk
Tk,χ,ξw2 w1 d − 1 2
k!
2.11
⎛
n n
j n−j m
⎝
w2 w1 En−j,χ,ξw1 w2 x
j
j0
×
j j
k0
k
Tk,χ,ξw2 w1 d −
m−1
1Ej−k,χ,ξw2 w1 y
tn
.
n!
m
In the viewpoint of the symmetry of Jχ,ξ w1 , w2 | x for w1 and w2 , we have
m
Jχ,ξ w1 , w2
| x ∞
n0
⎛
n n
j n−j m
⎝
w1 w2 En−j,χ,ξw2 w1 x
j
j0
×
j j
k0
k
m−1
Tk,χ,ξw1 w2 d − 1Ej−k,χ,ξw1 w2 y
2.12
n
t
.
n!
Comparing the coefficients on both sides of 2.11 and 2.12, we obtain the following
theorem.
Theorem 2.1. Let w1 , w2 , d ∈ N with w1 ≡ 1, w2 ≡ 1, and d ≡ 1 mod 2. For n ∈ N ∪ {0} and
m ∈ N, one has
j n n
j
j n−j m
m−1
w2 w1 En−j,χ,ξw1 w2 x
Tk,χ,ξw2 w1 d − 1Ej−k,χ,ξw2 w1 y
j
k
j0
k0
j n n
j
j n−j m
m−1
w1 w2 En−j,χ,ξw2 w1 x
Tk,χ,ξw1 w2 d − 1Ej−k,χ,ξw1 w2 y .
j
k
j0
k0
2.13
Let m 1 and y 0 in Theorem 2.1. Then we also have the following corollary.
Corollary 2.2. For w1 , w2 , d ∈ N with w1 ≡ 1, w2 ≡ 1, and d ≡ 1 mod 2, one has
n n
Em,χ,ξw1 w2 xw1m w2n−m Tn−m,χ,ξw2 w1 d − 1
m
m0
n n
Em,χ,ξw2 w1 xw1n−m w2m Tn−m,χ,ξw1 w2 d − 1
m
m0
2.14
see 2.
6
Journal of Inequalities and Applications
Let χ be the trivial character and d 1. Then we also have the following corollary.
Corollary 2.3. Let w1 , w2 ∈ N with w1 ≡ 1, w2 ≡ 1 mod 2. Then one has
n n
j
j0
n−j
j
w1 w2 En−j,ξw1 w2 xTk,ξw2 w1 − 1
n n
j
j0
2.15
j n−j
w1 w2 En−j,ξw2 w1 xTk,ξw1 w2
− 1,
where En,ξ x are the nth twisted Euler polynomials.
If we take w2 1 in Corollary 2.3, then we obtain the following corollary.
Corollary 2.4 Distribution for the twisted Euler polynomials. For w1 ∈ N with w1 ≡ 1 mod
2, one has
En,ξ x n n
i
i0
w1i Ei,ξw1 xTn−i,ξ w1 − 1.
2.16
From 2.6, we can derive that
m
Jχ,ξ w1 , w2 | x
w d−1
1
χl−1 ξ
w2 l
Xm
l0
×ξ
m
i1
×
X m−1
e
k0
m
i1
χxi ξ
χxi i1
xi w2 /w1 lw2 xt
m−1
i1
xi w2 e
m−1
i1
dμx1 · · · dμxm xi w2 t
dμx1 · · · dμxm−1 2.17
i1
∞
n n
n0
xi w1 w1 m−1
m
l
k
×
m−1
w1k w2n−k En−k,χ,ξw2 w1 y
w
1 d−1
l w2 l
χl−1 ξ
l0
m
Ek,χ,ξw1
n
t
w2
.
l
w2 x w1
n!
m
By the symmetry property of Jχ,ξ w1 , w2 | x in w1 and w2 , we also see that
m
Jχ,ξ w1 , w2 | x
∞
n0
n n
k0
k
m−1
w2k w1n−k En−k,χ,ξw1 w2 y
w
2 d−1
l0
l w1 l
χl−1 ξ
m
Ek,χ,ξw2
w1
l
w1 x w2
tn
.
n!
2.18
Journal of Inequalities and Applications
7
Comparing the coefficients on both sides of 2.17 and 2.18, we obtain the following theorem
which shows the relationship between the power sums and the twisted generalized Euler
polynomials of higher order.
Theorem 2.5. Let w1 , w2 , d ∈ N with w1 ≡ 1, w2 ≡ 1, and d ≡ 1 mod 2. For n ∈ N ∪ {0} and
m ∈ N, one has
1 d−1
w
w2
m−1
m
χl−1l ξw2 l Ek,χ,ξw1 w2 x l
w1k w2n−k En−k,χ,ξw2 w1 y
k
w1
l0
n n
k0
n 2 d−1
w
w1
n
m−1
m
χl−1l ξw1 l Ek,χ,ξw2 w1 x l .
w2k w1n−k En−k,χ,ξw1 w2 y
k
w2
k0
l0
2.19
If we take x 0, y 0, and m 1 in Theorem 2.5, then we have the following identity:
n n
Ek,χ,ξw1 w1k w2n−k Tn−k,χ,ξw2 w1 d − 1
k
k0
n n
Ek,χ,ξw2 w2k w1n−k Tn−k,χ,ξw1 w2 d − 1.
k
k0
2.20
Acknowledgment
The present research has been conducted by the research grant of Kwangwoon University in
2010.
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