Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 164743, 8 pages
doi:10.1155/2009/164743
Research Article
On the Symmetric Properties for the Generalized
Twisted Bernoulli Polynomials
Taekyun Kim and Young-Hee Kim
Division of General Education-Mathematics, Kwangwoon University,
Seoul 139-701, South Korea
Correspondence should be addressed to Young-Hee Kim, yhkim@kw.ac.kr
Received 6 July 2009; Accepted 18 October 2009
Recommended by Narendra Kumar Govil
We study the symmetry for the generalized twisted Bernoulli polynomials and numbers. We give
some interesting identities of the power sums and the generalized twisted Bernoulli polynomials
using the symmetric properties for the p-adic invariant integral.
Copyright q 2009 T. Kim and Y.-H. Kim. 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
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
and Z N∪{0}. Let νp be the normalized exponential valuation of Cp with |p|p p−νp p p−1 .
Let UDZp be the space of uniformly differentiable function on Zp . For f ∈ UDZp ,
the p-adic invariant integral on Zp is defined as
I f p −1
1 fxdx lim N
fx
N →∞p
Zp
x0
N
1.1
see 1. From 1.1, we note that
I f1 I f f 0,
1.2
2
Journal of Inequalities and Applications
where f 0 dfx/dx|x0 and f1 x fx 1. For n ∈ N, let fn x fx n. Then we can
derive the following equation from 1.2:
n−1
I fn I f f i
1.3
i0
see 1–7.
Let d be a fixed positive integer. For n ∈ N, let
− Z/dpN Z,
X Xd lim←
X1 Zp ,
N
X∗ a dp Zp ,
1.4
0<a<dp
a,p1
a dpN Zp x ∈ X | x ≡ a mod dpN
,
where a ∈ Z lies in 0 ≤ a < dpN . It is easy to see that
fxdx X
Zp
fxdx,
for f ∈ UD Zp .
1.5
The ordinary Bernoulli polynomials Bn x are defined as
∞
tn
t
xt
,
e
B
x
n
n!
et − 1
n0
1.6
and the Bernoulli numbers Bn are defined as Bn Bn 0 see 1–19.
For n ∈ N, let Tp be the p-adic locally constant space defined by
Tp n≥1
Cpn lim Cpn ,
1.7
n→∞
n
where Cpn {ω | ωp 1} is the cyclic group of order pn . It is well known that the twisted
Bernoulli polynomials are defined as
∞
tn
t
xt
,
e
B
x
n,ξ
n!
ξet − 1
n0
ξ ∈ Tp ,
1.8
and the twisted Bernoulli numbers Bn,ξ are defined as Bn,ξ Bn,ξ 0 see 15–18.
Let χ be Dirichlet’s character with conductor d ∈ N. Then the generalized twisted
Bernoulli polynomials Bn,χ,ξ x attached to χ are defined as follows:
d−1
∞
χaξa eat t xt tn
,
e
B
x
n,χ,ξ
d dt
n!
a0 ξ e − 1
n0
ξ ∈ Tp .
1.9
Journal of Inequalities and Applications
3
The generalized twisted Bernoulli numbers attached to χ, Bn,χ,ξ , are defined as Bn,χ,ξ Bn,χ,ξ 0 see 16.
Recently, many authors have studied the symmetric properties of the p-adic invariant
integrals on Zp , which gave some interesting identities for the Bernoulli and the Euler
polynomials cf. 3, 6, 7, 13, 14, 20–27. The authors of this paper have established various
identities by the symmetric properties of the p-adic invariant integrals and investigated
interesting relationships between the power sums and the Bernoulli polynomials see
2, 3, 6, 7, 13.
The twisted Bernoulli polynomials and numbers and the twisted Euler polynomials
and numbers are very important in several fields of mathematics and physicscf. 15–18.
The second author has been interested in the twisted Euler numbers and polynomials and
the twisted Bernoulli polynomials and studied the symmetry of power sum and twisted
Bernoulli polynomials see 11–13.
The purpose of this paper is to study the symmetry for the generalized twisted
Bernoulli polynomials and numbers attached to χ. In Section 2, we give interesting identities
for the power sums and the generalized twisted Bernoulli polynomials using the symmetric
properties for the p-adic invariant integral.
2. Symmetry for the Generalized Twisted Bernoulli Polynomials
Let χ be Dirichlet’s character with conductor d ∈ N. For ξ ∈ Tp , we have
χxξx ext dx t
X
d−1
i it
i0 χiξ e
ξd edt − 1
∞
Bn,χ,ξ
n0
tn
,
n!
2.1
where Bn,χ,ξ are the nth generalized twisted Bernoulli numbers attached to χ. We also see that
the generalized twisted Bernoulli polynomials attached to χ are given by
t
χ y ξy exyt dy X
d−1
i it
i0 χiξ e
ext
ξd edt − 1
∞
n0
Bn,χ,ξ x
tn
.
n!
2.2
By 2.1 and 2.2, we see that
n
χ y ξy x y dy Bn,χ,ξ x.
χxξx xn dx Bn,χ,ξ ,
X
2.3
X
From 2.3, we derive that
Bn,χ,ξ x n
n
l0
l
Bl,χ,ξ xn−l .
2.4
4
Journal of Inequalities and Applications
By 1.5 and 2.3, we see that
d−1
1
χxξ e dx χaeat ξa
d a0
X
x xt
Zp
ξdx edxt dx d−1
dt
1
ea/ddt .
χaξa d dt
d a0
ξ e −1
2.5
From 2.2 and 2.5, we obtain that
χxξ e dx x xt
∞
X
d
n−1
n0
d−1
a
χaξ Bn,ξd
a
d
a0
tn
.
n!
2.6
Thus we have the following theorem from 2.1 and 2.6.
Theorem 2.1. For ξ ∈ Tp , one has
Bn,χ,ξ dn−1
d−1
a
.
χaξa Bn,ξd
d
a0
2.7
By 1.3 and 1.5, we have that for n ∈ N,
fx ndx X
fxdx X
n−1
f i,
2.8
i0
where f i dfx/dx|xi . Taking fx χxξx ext in 2.8, it follows that
1
t
χxξndx endxt dx −
X
χx ξx ext dx
X
d−1
nd χx ξx ext dx ξnd endt − 1
d dt
X ndx ndxt
ξ e −1
ξ e
dx
X
2.9
i it
χiξ e
.
i0
Thus, we have
1
t
χxξndx endxt dx −
χxξx ext dx
X
X
∞
nd−1
k0
l0
χl ξl lk
tk
.
k!
2.10
For k ∈ Z , let us define the p-adic functional Kχ, ξ, k : n as follows:
n
K χ, ξ, k : n χlξl lk .
2.11
l0
By 2.10 and 2.11, we see that for k, n, d ∈ N,
χxξ
X
ndx
χxξx xk dx kK χ, ξ, k − 1 : nd − 1 .
k
nd x dx −
X
2.12
Journal of Inequalities and Applications
5
From 2.3 and 2.12, we have the following result.
Theorem 2.2. For ξ ∈ Tp and k, n, d ∈ N, one has
ξnd Bk,χ,ξ nd − Bk,χ,ξ kK χ, ξ, k − 1 : nd − 1 .
2.13
Let w1 , w2 , d ∈ N. Then we have that
d
χx1 χx2 ξw1 x1 w2 x2 ew1 x1 w2 x2 t dx1 dx2
ξdw1 w2 x edw1 w2 xt dx
X
d−1
t ξdw1 w2 edw1 w2 t − 1
w d dw t
χaξw1 a ew1 at
w d dw t
ξ 1 e 1 −1 ξ 2 e 2 −1
a0
X X
d−1
χbξw2 b ew2 bt .
2.14
b0
By 2.9, 2.10, and 2.11, we see that
∞
w1 d X χxξx ext dx tk
.
K
χ,
ξ,
k
:
dw
−
1
1
k!
ξdw1 x edw1 xt dx
k0
X
2.15
Now let us define the p-adic functional Yχ,ξ w1 , w2 as follows:
Yχ,ξ w1 , w2 d
X X
χx1 χx2 ξw1 x1 w2 x2 ew1 x1 w2 x2 w1 w2 xt dx1 dx2
.
ξdw1 w2 x3 edw1 w2 x3 t dx3
X
2.16
Then it follows from 2.14 that
t ξdw1 w2 edw1 w2 t − 1 ew1 w2 xt
Yχ,ξ w1 , w2 w d dw t
ξ 1 e 1 − 1 ξw2 d edw2 t − 1
d−1
χaξw1 a ew1 at
a0
d−1
χbξw2 b ew2 bt .
b0
2.17
By 2.15 and 2.16, we obtain that
Yχ,ξ w1 , w2 1
w1
χx1 ξ
e
X
i0
dx1
dw1
X
∞
l
l
l0
w1 x1 w1 x1 w2 xt
i
χx2 ξw2 x2 ew2 x2 t dx2
X
ξdw1 w2 x edw1 w2 xt dx
Bi,χ,ξw1 w2 xK χ, ξw2 , l − i : dw1 − 1 w1i−1 w2l−i
tl
.
l!
2.18
6
Journal of Inequalities and Applications
On the other hand, the symmetric property of Yχ,ξ w1 , w2 shows that
Yχ,ξ w1 , w2 1
w2
χx2 ξ
e
dw2
dx2
X
i0
X
∞
l
l
l0
w2 x2 w2 x2 w1 xt
χx1 ξw1 x1 ew1 x1 t dx1
X
ξdw1 w2 x edw1 w2 xt dx
Bi,χ,ξw2 w1 xK χ, ξ , l − i : dw2 − 1 w2i−1 w1l−i
i
w1
tl
.
l!
2.19
Comparing the coefficients on the both sides of 2.18 and 2.19, we have the following
theorem.
Theorem 2.3. Let ξ ∈ Tp and d, w1 , w2 ∈ N. Then one has
l
l
i
i0
Bi,χ,ξw1 w2 xK χ, ξw2 , l − i : dw1 − 1 w1i−1 w2l−i
l
l
i0
Bi,χ,ξw2 w1 xK χ, ξw1 , l − i : dw2 − 1 w2i−1 w1l−i .
i
2.20
We also derive some identities for the generalized twisted Bernoulli numbers. Taking
x 0 in Theorem 2.3, we have the following corollary.
Corollary 2.4. Let ξ ∈ Tp and d, w1 , w2 ∈ N. Then one has
l
l
i0
l
l
Bi,χ,ξw1 K χ, ξw2 , l−i : dw1 −1 w1i−1 w2l−i
Bi,χ,ξw2 K χ, ξw1 , l − i : dw2 − 1 w2i−1 w1l−i .
i
i0 i
2.21
Now we will derive another identities for the generalized twisted Bernoulli
polynomials using the symmetric property of Yχ,ξ w1 , w2 . From 1.2, 2.15 and 2.17, we
see that
Yχ,ξ w1 , w2 ew1 w2 xt
w1
χx1 ξ
X
1 −1
1 dw
χiξw2 i
w1 i0
w 1 x1 w 1 x1 t
e
dx1
dw1
X
χx2 ξw2 x2 ew2 x2 t dx2
X
ξdw1 w2 x edw1 w2 xt dx
χx1 ξw1 x1 ew1 x1 w2 xw2 /w1 it dx1
X
w2
χiξw2 i Bk,χ,ξw1 w2 x i w1k−1
w1
∞
dw
1 −1
k0
i0
tk
.
k!
2.22
Journal of Inequalities and Applications
7
From the symmetric property of Yχ,ξ w1 , w2 , we also see that
Yχ,ξ w1 , w2 ew1 w2 xt
w2
χx2 ξ
w 2 x2 w 2 x2 t
e
dx2
dw2
X
X
2 −1
1 dw
χiξw1 i
w2 i0
∞
dw
2 −1
k0
i0
χx2 ξw2 x2 ew2 x2 w1 xw1 /w2 it dx2
2.23
X
χiξ
χx1 ξw1 x1 ew1 x1 t dx1
X
ξdw1 w2 x edw1 w2 xt dx
w1 i
Bk,χ,ξw2
w1
i w2k−1
w1 x w2
tk
.
k!
Comparing the coefficients on the both sides of 2.22 and 2.23, we obtain the following
theorem.
Theorem 2.5. Let ξ ∈ Tp and d, w1 , w2 ∈ N. Then one has
dw
1 −1
χiξ
w2 i
B
k,χ,ξw1
i0
dw
2 −1
w2
w1
k−1
w1 i
w
i w1 χiξ Bk,χ,ξ 2 w1 x i w2k−1 .
w2 x w1
w2
i0
2.24
If we take x 0 in Theorem 2.5, we also derive the interesting identity for the
generalized twisted Bernoulli numbers as follows: for d, w1 , w2 ∈ N,
dw
1 −1
χiξ
i0
w2 i
B
k,χ,ξw1
dw
2 −1
w2
w1
k−1
w1 i
w
i w1 χiξ Bk,χ,ξ 2
i w2k−1 .
w1
w2
i0
2.25
Acknowledgment
The present research has been conducted by the research grant of the Kwangwoon University
in 2009.
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