Volume 2009, Article ID 640152,7pages doi:10.1155/2009/640152
Research Article
On the Identities of Symmetry for
the Generalized Bernoulli Polynomials
Attached to
χ
of Higher Order
Taekyun Kim,
1Lee-Chae Jang,
2Young-Hee Kim,
1and Kyung-Won Hwang
31Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea 2Department of Mathematics and Computer Science, Konkook University, Chungju 139-701, South Korea 3Department of General Education, Kookmin University, Seoul 136-702, South Korea
Correspondence should be addressed to Taekyun Kim,[email protected]
Received 5 June 2009; Accepted 5 August 2009
Recommended by Vijay Gupta
We give some interesting relationships between the power sums and the generalized Bernoulli numbers attached toχof higher order using multivariatep-adic invariant integral onZp.
Copyrightq2009 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.
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 ofp-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, andZ N∪ {0}. Letνp be the normalized exponential valuation ofCp
with |p|p p−νpp p−1 see1–24. Let UDZpbe the space of uniformly differentiable function onZp. Letdbe a fixed positive integer. Forn∈N, let
XXdlim
← N
Z
dpNZ, X1Zp,
X∗ 0<a<dp
a,p1
adpZp
,
adpNZ
p
x∈X|x≡amoddpN,
wherea∈Zlies in 0≤a < dpN. Forf∈UDX, thep-adic invariant integral onXis defined as
If
X
fxdx lim N→ ∞
1
dpN dpN−1
x0
fx 1.2
see11–19. From1.2, we note that
If1
Iff0, 1.3
wheref0 dfx/dx|x0andf1x fx1. Letfnx fxn n∈N. Then we can
derive the following equation from1.3:
Ifn
If
n−1
i0
fi 1.4
see1–11. Letχbe the Dirichlet’s character with conductord ∈ N. Then the generalized
Bernoulli polynomials attached toχare defined as
d−1
a0
χaeatt
edt−1 e xt∞
n0
Bn,χxt n
n!, 1.5
and the generalized Bernoulli numbers attached to χ,Bn,χ, are defined as Bn,χ Bn,χ0
see1–20,25. The purpose of this paper is to derive some identities of symmetry for the
generalized Bernoulli polynomials attached toχof higher order.
2. Symmetric Properties for the Generalized Bernoulli
Polynomials of Higher Order
Letχbe the Dirichlet’s character with conductord∈N. Then we note that
X
χxextdx t
d−1 i0 χieit edt−1
∞
n0 Bn,χt
n
n!, 2.1
whereBn,χare thenth generalized Bernoulli numbers attached toχsee7,9,15,25. Now
we also see that the generalized Bernoulli polynomials attached toχare given by
X
χyexytdy t d−1
i0 χieit edt−1 e
xt∞
n0
Bn,χxt n
n!. 2.2
see15,25, and
X
χyxyndyBn,χx 2.4
see1–19,25. Forn∈N, we obtain that
X
fxndx
X
fxdx
n−1
i0
fi, 2.5
wherefi dfx/dx|xi. Thus, we have
1
t X
χxendxtdx−
X
χxextdx
nd
Xχxextdx
Xendxtdx
endt−1
edt−1 d−1
i0 χieit
. 2.6
Then
1
t X
χxendxtdx− X
χxextdx
nd−1
l0
χlelt∞ k0
nd−1
l0 χllk
tk
k!. 2.7
Let us define thep-adic functionTkχ, nas follows:
Tk
χ, n
n
l0
χllk 2.8
see25. By2.7and2.8, we see that
1
t X
χxendxtdx− X
χxextdx
∞
k0 Tk
χ, nd−1t k
k! 2.9
see25. Thus, we have
X
χxndxkdx−
X
χxxkdxkTk−1
χ, nd−1, k, n, d∈N. 2.10
This means that
Bk,χnd−Bk,χkTk−1
χ, nd−1, k, n, d∈N 2.11
The generalized Bernoulli polynomials attached toχof orderk, which is denoted by
Bn,χkx, are defined as
tdi−01χieit
edt−1 k
ext
∞
n0
Bn,χkxt n
n!. 2.12
Then the values ofBn,χkxatx0 are called the generalized Bernoulli numbers attached to
χof orderk. Whenk 1, the polynomials of numbers are called the generalized Bernoulli
polynomials or numbers attached toχ. Letw1, w2 ∈N. Then we set
Km, χ;w1, w2
d
Xm
m
i1χxie
m
i1xiw2xw1tm
i1dxi
Xm
m
i1χxie
m
i1xiw1yw2tm
i1dxi
Xedw1w2xtdx
,
2.13
where
Xm
fx1, . . . , xmdx1· · ·dxm X
· · ·
X
fx1, . . . , xmdx1· · ·dxm. 2.14
In2.13, we note thatKm, χ;w1, w2is symmetric inw1, w2. From2.13, we derive
Km, χ;w1, w2
Xm m
i1
χxie
m
i1xiw1tdx
1· · ·dxm
ew1w2xt
dXχxmew2xmtdxm
Xedw1w2xtdx
×
Xm−1
m−1
i1
χxie
m−1
i1 xiw2tdx
1· · ·dxm−1
ew1w2yt.
2.15
It is easy to see that
w1d
Xχxextdx
Xedw1xtdx
∞
k0
w
1d−1
i0 χiik
tk
k!
∞
k0 Tk
χ, w1d−1
tk
k!,
ew1w2xt
Xm m
i1
χxie
m
i1xiw1tdx
1· · ·dxm
w t d−1 m ∞ tn
From2.16, we note that
Km, χ;w1, w2
∞
l0
Bl,χmw2xw1
ltl
l!
∞
k0 Tk
χ, w1d−1
w2ktk k!
∞
i0
Bi,χm−1w1y
w2iti i!
1
w1
∞
n0
⎡ ⎣n
j0
n j
w2jwn1−j−1Bnm−j,χ w2x
j
k0 Tk
χ, w1d−1
j
k
Bjm−k,χ−1w1y
⎤⎦tn
n!.
2.17
By the symmetry ofKm, χ;w1, w2inw1andw2, we see that
Km, χ;w1, w2
∞
n0
⎡ ⎣n
j0
n j
w1jwn2−j−1Bnm−j,χ w1x
j
k0 Tk
χ, w2d−1
j
k
Bj−mk,χ−1w2y
⎤⎦tn n!.
2.18
By comparing the coefficients on the both sides of 2.17and 2.18, we see the following
theorem.
Theorem 2.1. Ford, w1, w2∈N, n≥0, m≥1, one has
n
j0
n j
w2jwn1−j−1Bnm−j,χ w2x
j
k0 Tk
χ, w1d−1
j
k
Bjm−k,χ−1w1y
n
j0
n j
w1jwn2−j−1Bnm−j,χ w1x
j
k0 Tk
χ, w2d−1
j
k
Bjm−k,χ−1w2y
.
2.19
Remark 2.2. Lety0 andm1 in1.4. Then we have
n
j0
n j
wj2wn1−j−1Bn−j,χw2xTj
χ, w1d−1
n
j0
n j
wj1wn2−j−1Bn−j,χw1xTj
χ, w2d−1
2.20
We also calculate that
Km, χ;w1, w2
∞
n0
n
k0
n k
w1k−1w2n−kBnm−k,χ−1w1y
dw1−1
i0 Bk,χm
w2xw2 w1i
tn
n!.
2.21
From the symmetric property ofKm, χ;w1, w2inw1andw2, we derive
Km, χ;w1, w2
∞
n0
n
k0
n k
w2k−1w1n−kBnm−k,χ−1w2y
dw2−1
i0 Bk,χm
w1xw1 w2i
tn n!.
2.22
By comparing the coefficients on the both sides of2.21and2.22, we obtain the following
theorem.
Theorem 2.3. Forw1, w2∈N, n∈Z, m∈N, one has
n
k0
n k
wk−1 1 w2n−kB
m−1
n−k,χ
w1y
dw1−1
i0 Bk,χm
w2xw2 w1i
n
k0
n k
wk2−1wn1−kB m−1
n−k,χ
w2y
dw2−1
i0 Bk,χm
w1xw1 w2i
.
2.23
Remark 2.4. Lety0 andm1 in2.23. We have
w1n−1
dw1−1
i0 Bn,χ
w2xw2 w1i
wn2−1
dw2−1
i0 Bn,χ
w1xw1 w2i
2.24
see25.
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