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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
Z
p
Associated with the Twisted Generalized Euler
Polynomials of H igher Order
Taekyun Kim,
1
Byungje Lee,
2
and Young-Hee Kim
1
1
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2
Department of Wireless Communications Engineering, Kwangwoon University,
Seoul 139-701, Republic of Korea
Correspondence should be addressed to Young-Hee Kim,
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
Z
p
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, Z
p
, Q
p
,andC
p
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 Q
p
, respectively. Let N be the set of natural
numbers. The normalized valuation in C
p
is denoted by |·|
p
with |p|
p
1/p.LetUDZ
p
be the space of uniformly differentiable function on Z
p
. For f ∈ UDZ
p
, the fermionic p-adic
invariant integral on Z
p
is defined as
I
f
Z
p
f
x
dμ
x
lim
N →∞
p
N
−1
x0
f
x
μ
x p
N
Z
p
lim
N →∞
p
N
−1
x0
f
x
−1
x
1.1
see 1–25. For n ∈ N,wenotethat
Z
p
f
x n
dμ
x
−1
n
Z
p
f
x
dμ
x
n−1
l0
−1
n−1−l
f
l
see
5
. 1.2
2 Journal of Inequalities and Applications
Let d be a fixed odd positive integer. For N ∈ N,weset
X X
d
lim
←−
N
Z/dp
N
Z,X
1
Z
p
,
X
∗
0<a<dp
a,p1
a dpZ
p
,
a dp
N
Z
p
x ∈ X | x ≡ a
mod dp
N
,
1.3
where a ∈ Z lies in 0 ≤ a<dp
N
see 1–13. It is well known that for f ∈ UDZ
p
,
X
f
x
dμ
x
Z
p
f
x
dμ
x
. 1.4
For n ∈ N,letC
p
n
be the cyclic group of order p
n
.Thatis,C
p
n
{ξ | ξ
p
n
1}.Thep-adic
locally constant space, T
p
, is defined by T
p
lim
n →∞
C
p
n
n≥1
C
p
n
.
Let χ be Dirichlet’s character with conductor d ∈ N and let ξ ∈ T
p
. Then the generalized
twisted Bernoulli polynomials B
n,χ,ξ
x attached to χ are defined as
t
d−1
a0
χ
a
ξ
a
e
at
ξ
d
e
dt
− 1
e
xt
∞
n0
B
n,χ,ξ
x
t
n
n!
see
10
. 1.5
In 4, 7, 10–12, t he generalized twisted Bernoulli polynomials of order k attached to χ
are also defined as follows:
t
d−1
a0
χ
a
ξ
a
e
at
ξ
d
e
dt
− 1
×···×
t
d−1
a0
χ
a
ξ
a
e
at
ξ
d
e
dt
− 1
k-times
e
xt
∞
n0
B
k
n,χ,ξ
x
t
n
n!
. 1.6
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 Z
p
. 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 Z
p
.
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.Thatis,d ∈ N with d ≡ 1 mod 2.
For ξ ∈ T
p
, the twisted generalized Euler polynomials attached to χ, E
n,χ,ξ
x, are defined as
X
χ
y
ξ
y
e
xyt
dμ
y
2
d−1
a0
−1
a
χ
a
ξ
a
e
at
ξ
d
e
dt
1
e
xt
∞
n0
E
n,χ,ξ
x
t
n
n!
see
12
. 2.1
In the special case x 0, E
n,χ,ξ
E
n,χ,ξ
0 are called the nth twisted generalized Euler numbers
attached to χ.
From 2.1,wenotethat
X
χ
y
ξ
y
x y
m
dμ
y
E
m,χ,ξ
x
,m∈ N ∪
{
0
}
. 2.2
For n ∈ N with n ≡ 1 mod 2, we have
X
χ
x
ξ
x
e
xndt
dμ
x
X
χ
x
ξ
x
e
xt
dμ
x
2
nd−1
l0
−1
l
χ
l
ξ
l
e
lt
. 2.3
Let T
k,χ,ξ
n
n
l0
−1
l
χlξ
l
l
k
. Then we see that
ξ
nd
X
χ
x
ξ
x
e
xndt
dμ
x
X
χ
x
ξ
x
e
xt
dμ
x
2
X
e
xt
χ
x
ξ
x
dμ
x
X
e
ndxt
ξ
ndx
dμ
x
2
∞
k0
T
k,χ,ξ
nd − 1
t
k
k!
.
2.4
Now we define the twisted generalized Euler polynomials E
k
n,χ,ξ
x of order k attached
to χ as follows:
e
xt
2
d−1
a0
−1
a
χaξ
a
e
at
ξ
d
e
dt
1
k
∞
n0
E
k
n,χ,ξ
x
t
n
n!
. 2.5
In the special case x 0, E
k
n,χ,ξ
E
k
n,χ,ξ
0 are called the nth twisted generalized Euler numbers
of order k.
4 Journal of Inequalities and Applications
Let w
1
,w
2
,d ∈ N with w
1
≡ 1,w
2
≡ 1, and d ≡ 1 mod 2. Then we set
J
m
χ,ξ
w
1
,w
2
| x
X
m
m
i1
χ
x
i
ξ
m
i1
x
i
w
1
e
m
i1
x
i
w
2
xw
1
t
dμ
x
1
···dμ
x
m
X
ξ
dw
1
w
2
x
e
dw
1
w
2
xt
dμ
x
×
X
m
m
i1
χ
x
i
ξ
m
i1
x
i
w
2
e
m
i1
x
i
w
1
yw
2
t
dμ
x
1
···dμ
x
m
,
2.6
where
X
m
f
x
1
, ,x
m
dμ
x
1
···dμ
x
m
X
···
X
m-times
f
x
1
, ,x
m
dμ
x
1
···dμ
x
m
. 2.7
From 2.6,wenotethat
J
m
χ,ξ
w
1
,w
2
| x
X
m
m
i1
χ
x
i
ξ
m
i1
x
i
w
1
e
m
i1
x
i
w
1
t
dμ
x
1
···dμ
x
m
e
w
1
w
2
xt
×
X
χ
x
m
ξ
w
2
x
m
e
w
2
x
m
t
dμ
x
m
X
ξ
dw
1
w
2
x
e
dw
1
w
2
xt
dμ
x
e
w
1
w
2
yt
×
X
m−1
m−1
i1
χ
x
i
ξ
m−1
i1
x
i
w
2
e
m−1
i1
x
i
w
2
t
dμ
x
1
···dμ
x
m−1
.
2.8
From 2.4, we can easily derive the following equation:
X
χ
x
ξ
x
e
xt
dμ
x
X
ξ
dw
1
x
e
dw
1
xt
dμ
x
dw
1
−1
l0
−1
l
χ
l
ξ
l
e
lt
∞
k0
T
k,χ,ξ
dw
1
− 1
t
k
k!
. 2.9
It is not difficult to show that
e
w
1
w
2
xt
X
m
m
i1
χ
x
i
ξ
m
i1
x
i
w
1
e
m
i1
x
i
w
1
t
dμ
x
1
···dμ
x
m
2
d−1
a0
−1
a
χaξ
aw
1
e
aw
1
t
ξ
dw
1
e
dw
1
t
1
m
e
w
1
w
2
xt
∞
k0
E
m
k,χ,ξ
w
1
w
2
x
w
k
1
t
k
k!
.
2.10
Journal of Inequalities and Applications 5
By 2.8, 2.9,and2.10,weseethat
J
m
χ,ξ
w
1
,w
2
| x
∞
l0
E
m
l,χ,ξ
w
1
w
2
x
w
l
1
t
l
l!
∞
k0
T
k,χ,ξ
w
2
w
1
d − 1
w
k
2
t
k
k!
×
∞
i0
E
m−1
i,χ,ξ
w
2
w
1
y
w
i
2
t
i
i!
∞
n0
⎛
⎝
n
j0
n
j
w
j
2
w
n−j
1
E
m
n−j,χ,ξ
w
1
w
2
x
×
j
k0
j
k
T
k,χ,ξ
w
2
w
1
d − 1
E
m−1
j−k,χ,ξ
w
2
w
1
y
t
n
n!
.
2.11
In the viewpoint of the symmetry of J
m
χ,ξ
w
1
,w
2
| x for w
1
and w
2
, we have
J
m
χ,ξ
w
1
,w
2
| x
∞
n0
⎛
⎝
n
j0
n
j
w
j
1
w
n−j
2
E
m
n−j,χ,ξ
w
2
w
1
x
×
j
k0
j
k
T
k,χ,ξ
w
1
w
2
d − 1
E
m−1
j−k,χ,ξ
w
1
w
2
y
t
n
n!
.
2.12
Comparing the coefficients on both sides of 2.11 and 2.12, we obtain the following
theorem.
Theorem 2.1. Let w
1
,w
2
,d ∈ N with w
1
≡ 1,w
2
≡ 1, and d ≡ 1 mod 2. For n ∈ N ∪{0} and
m ∈ N, one has
n
j0
n
j
w
j
2
w
n−j
1
E
m
n−j,χ,ξ
w
1
w
2
x
j
k0
j
k
T
k,χ,ξ
w
2
w
1
d − 1
E
m−1
j−k,χ,ξ
w
2
w
1
y
n
j0
n
j
w
j
1
w
n−j
2
E
m
n−j,χ,ξ
w
2
w
1
x
j
k0
j
k
T
k,χ,ξ
w
1
w
2
d − 1
E
m−1
j−k,χ,ξ
w
1
w
2
y
.
2.13
Let m 1andy 0inTheorem 2.1. Then we also have the following corollary.
Corollary 2.2. For w
1
,w
2
,d ∈ N with w
1
≡ 1,w
2
≡ 1, and d ≡ 1 mod 2, one has
n
m0
n
m
E
m,χ,ξ
w
1
w
2
x
w
m
1
w
n−m
2
T
n−m,χ,ξ
w
2
w
1
d − 1
n
m0
n
m
E
m,χ,ξ
w
2
w
1
x
w
n−m
1
w
m
2
T
n−m,χ,ξ
w
1
w
2
d − 1
see
2
.
2.14
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 w
1
,w
2
∈ N with w
1
≡ 1,w
2
≡ 1 mod 2. Then one has
n
j0
n
j
w
n−j
1
w
j
2
E
n−j,ξ
w
1
w
2
x
T
k,ξ
w
2
w
1
− 1
n
j0
n
j
w
j
1
w
n−j
2
E
n−j,ξ
w
2
w
1
x
T
k,ξ
w
1
w
2
− 1
,
2.15
where E
n,ξ
x are the nth twisted Euler polynomials.
If we take w
2
1inCorollary 2.3, then we obtain the following corollary.
Corollary 2.4 Distribution for the twisted Euler polynomials. For w
1
∈ N with w
1
≡ 1 mod
2, one has
E
n,ξ
x
n
i0
n
i
w
i
1
E
i,ξ
w
1
x
T
n−i,ξ
w
1
− 1
. 2.16
From 2.6, we can derive that
J
m
χ,ξ
w
1
,w
2
| x
w
1
d−1
l0
χ
l
−1
l
ξ
w
2
l
X
m
m
i1
χ
x
i
×ξ
m
i1
x
i
w
1
e
w
1
m
i1
x
i
w
2
/w
1
lw
2
xt
dμ
x
1
···dμ
x
m
×
X
m−1
m−1
i1
χ
x
i
ξ
m−1
i1
x
i
w
2
e
m−1
i1
x
i
w
2
t
dμ
x
1
···dμ
x
m−1
∞
n0
n
k0
n
k
w
k
1
w
n−k
2
E
m−1
n−k,χ,ξ
w
2
w
1
y
×
w
1
d−1
l0
χ
l
−1
l
ξ
w
2
l
E
m
k,χ,ξ
w
1
w
2
x
w
2
w
1
l
t
n
n!
.
2.17
By the symmetry property of J
m
χ,ξ
w
1
,w
2
| x in w
1
and w
2
,wealsoseethat
J
m
χ,ξ
w
1
,w
2
| x
∞
n0
n
k0
n
k
w
k
2
w
n−k
1
E
m−1
n−k,χ,ξ
w
1
w
2
y
w
2
d−1
l0
χ
l
−1
l
ξ
w
1
l
E
m
k,χ,ξ
w
2
w
1
x
w
1
w
2
l
t
n
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 t he relationship between the power sums and the twisted generalized Euler
polynomials of higher order.
Theorem 2.5. Let w
1
,w
2
,d ∈ N with w
1
≡ 1,w
2
≡ 1, and d ≡ 1 mod 2. For n ∈ N ∪{0} and
m ∈ N, one has
n
k0
n
k
w
k
1
w
n−k
2
E
m−1
n−k,χ,ξ
w
2
w
1
y
w
1
d−1
l0
χ
l
−1
l
ξ
w
2
l
E
m
k,χ,ξ
w
1
w
2
x
w
2
w
1
l
n
k0
n
k
w
k
2
w
n−k
1
E
m−1
n−k,χ,ξ
w
1
w
2
y
w
2
d−1
l0
χ
l
−1
l
ξ
w
1
l
E
m
k,χ,ξ
w
2
w
1
x
w
1
w
2
l
.
2.19
If we take x 0, y 0, and m 1inTheorem 2.5, then we have the following identity:
n
k0
n
k
E
k,χ,ξ
w
1
w
k
1
w
n−k
2
T
n−k,χ,ξ
w
2
w
1
d − 1
n
k0
n
k
E
k,χ,ξ
w
2
w
k
2
w
n−k
1
T
n−k,χ,ξ
w
1
w
2
d − 1
.
2.20
Acknowledgment
The present research has been conducted by the research grant of Kwangwoon University in
2010.
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