Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2009, Article ID 848943, 8 pages
doi:10.1155/2009/848943
Research Article
Some Identities of Symmetry for the Generalized
Bernoulli Numbers and Polynomials
Taekyun Kim,1 Seog-Hoon Rim,2 and Byungje Lee3
1
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
Department of Mathematics Education, Kyungpook National University, Taegu 702-701, South Korea
3
Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701,
South Korea
2
Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr
Received 4 April 2009; Accepted 11 May 2009
Recommended by Lance Littlejohn
By the properties of p-adic invariant integral on Zp , we establish various identities concerning
the generalized Bernoulli numbers and polynomials. From the symmetric properties of p-adic
invariant integral on Zp , we give some interesting relationship between the power sums and the
generalized Bernoulli polynomials.
Copyright q 2009 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 will
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 vp be the normalized exponential valuation of Cp
with |p|p p−vp p 1/p. 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 the definition 1.1, we have
I f1 I f f 0,
dfx , f1 x fx 1.
where f 0 dx x0
1.2
2
Abstract and Applied Analysis
Let fn x fx n, n ∈ N. Then we can derive the following equation from 1.2:
n−1
I fn I f f i,
1.3
i0
see 1. It is well known that the ordinary Bernoulli polynomials Bn x are defined as
et
∞
tn
t
ext Bn x ,
n!
−1
n0
1.4
see 1–25, and the Bernoulli number Bn are defined as Bn Bn 0.
Let d be a fixed positive integer. For n ∈ N, we set
X Xd lim Z/dpN Z ,
←
N
X∗ X1 Zp ;
a dpZp ;
0<a<dp,
a,p1
1.5
a dpN Zp x ∈ X | x ≡ a mod dpN ,
where a ∈ Z lies in 0 ≤ a < dpN . It is easy to see that
fxdx Zp
X
fxdx,
for f ∈ UD Zp .
1.6
In 14, the Witt’s formula for the Bernoulli numbers are given by
Zp
xn dx Bn ,
n ∈ Z .
1.7
Let χ be the Dirichlet’s character with conductor d ∈ N. Then the generalized Bernoulli
polynomials attached to χ are defined as
d
χateat
a1
edt
−1
ext ∞
tn
Bn,χ x ,
n!
n0
1.8
see 22, and the generalized Bernoulli numbers attached to χ, Bn,χ are defined as Bn,χ Bn,χ 0.
In this paper, we investigate the interesting identities of symmetry for the generalized
Bernoulli numbers and polynomials attached to χ by using the properties of p-adic invariant
integral on Zp . Finally, we will give relationship between the power sum polynomials and the
generalized Bernoulli numbers attached to χ.
Abstract and Applied Analysis
3
2. Symmetry of Power Sum and the Generalized Bernoulli Polynomials
Let χ be the Dirichlet character with conductor d ∈ N. From 1.3, we note that
t
χxe dx xt
d−1
i0 χie
edt − 1
X
it
∞
tn
Bn,χ ,
n!
n0
2.1
where Bn,χ x are the nth generalized Bernoulli numbers attached to χ. Now, we also see that
the generalized Bernoulli polynomials attached to χ are given by
t
χ y exyt dy d−1
i0 χie
edt − 1
X
it
ext ∞
tn
Bn,χ x .
n!
n0
2.2
By 2.1 and 2.2, we easily see that
χ y x yn dy Bn,χ x.
χxx dx Bn,χ ,
n
X
2.3
X
From 2.2, we have
Bn,χ x n n
B,χ xn− .
0
2.4
From 2.2, we can also derive
n
∞
d−1
t
t
i
i/ddt
n−1
d
.
e
χxe dx χi dt
χiBn
d
n!
e −1
X
n0
i0
i0
d−1
xt
2.5
Therefore, we obtain the following lemma.
Lemma 2.1. For n ∈ Z , one has
χxxn dx Bn,χ dn−1
X
d−1
χiBi
i0
i
.
d
2.6
We observe that
1
t
χxe
ndxt
dx −
X
xt
e χxdx
X
nd
X
χxext dx
X
endxt dx
endt − 1
dt
e −1
d−1
χie
it
.
2.7
i0
Thus, we have
1
t
χxe
X
ndxt
dx −
xt
χxe dx
X
∞
k0
nd−1
χ
0
k
tk
.
k!
2.8
4
Abstract and Applied Analysis
Let us define the p-adic functional Tk χ, n as follows:
n
χ k ,
Tk χ, n for k ∈ Z .
2.9
0
By 2.8 and 2.9, we see that
1
t
χxendxt dx −
X
χxext dx
X
∞
n0
tk
.
Tk χ, nd − 1
k!
2.10
By using Taylor expansion in 2.10, we have
χxxk dx kTk−1 χ, nd − 1 ,
χxdn xk dx −
X
for k, n, d ∈ N .
2.11
X
That is,
Bk,χ nd − Bk,χ kTk−1 χ, nd − 1 .
2.12
Let w1 , w2 , d ∈ N. Then we consider the following integral equation:
d
X
d−1
d−1
χx1 χx2 ew1 x1 w2 x2 t dx1 dx2
t edw1 w2 t − 1
w1 at
w2 bt
.
w dt
χae
χbe
edw1 w2 xt dx
e 1 − 1 ew2 dt − 1
a0
b0
X
2.13
From 2.7 and 2.10, we note that
dw1
X
X
χxext dx
edw1 xt dx
∞
k0
tk
.
Tk χ, dw1 − 1
k!
2.14
Let us consider the p-adic functional Tχ w1 , w2 as follows:
Tχ w1 , w2 d
X
χx1 χx2 ew1 x1 w2 x2 w1 w2 xt dx1 dx2
.
edw1 w2 x3 t dx3
X
2.15
Then we see that Tχ w1 , w2 is symmetric in w1 and w2 , and
d−1
d−1
t edw1 w2 t − 1 ew1 w2 xt w1 at
w2 bt
.
Tχ w1 , w2 w dt
χae
χbe
e 1 − 1 ew2 dt − 1
a0
b0
2.16
Abstract and Applied Analysis
5
By 2.15 and 2.16, we have
dw1 X χx2 ew2 x2 t dx2
1
w1 x1 w2 xt
χx1 e
dx1
Tχ w1 , w2 w1 X
edw1 w2 xt dx
X
∞
∞
w1i ti
w2k tk
1 Bi,χ w2 x
Tk χ, dw1 − 1
w1 i0
i!
k!
k0
1
w1
i −i ∞
B w xT
t
i,χ
2
−i χ, dw1 − 1 w1 w2 !
0
i0
i! − i!
2.17
!
∞
i−1 −i t
.
Bi,χ w2 xT−i χ, dw1 − 1 w1 w2
i
!
i0
0
From the symmetric property of Tχ w1 , w2 in w1 and w2 , we note that
dw2 X χx1 ew1 x1 t dx1
1
w2 x2 w1 xt
Tχ w1 , w2 χx2 e
dx2
w2 X
edw1 w2 xt dx
X
∞
∞
w2i ti
w1k tk
1 Bi,χ w1 x
Tk χ, dw2 − 1
w2 i0
i!
k!
k0
1
w2
−i ∞
B w xw i T
t
i,χ
1
2 −i χ, dw2 − 1 w1 !
0
i0
i! − i!
2.18
!
∞
t
i−1 −i
.
w2 w1 Bi,χ w1 xT−i χ, dw2 − 1
i
!
i0
0
By comparing the coefficients on the both sides of 2.17 and 2.18, we obtain the following
theorem.
Theorem 2.2. For w1 , w2 , d ∈ N, one has
Bi,χ w2 xT−i χ, dw1 − 1 w1i−1 w2−i Bi,χ w1 xT−i χ, dw2 − 1 w2i−1 w1−i .
i
i
i0
i0
2.19
Let x 0 in Theorem 2.2. Then we have
i0
Bi,χ T−i χ, dw1 − 1 w1i−1 w2−i Bi,χ T−i χ, dw2 − 1 w2i−1 w1−i .
i
i
i0
2.20
6
Abstract and Applied Analysis
By 2.14 and 2.16, we also see that
Tχ w1 , w2 ew1 w2 xt
w1
χx1 e
w 1 x1 t
dx1
dw1
X
χx2 ew2 x2 t dx2
edw1 w2 xt dx
w1 w2 xt dw1 w2 t
d−1
e
e
−1
w 1 x1 t
w2 it
χx1 e
dx1
χie
w1
ew2 dt − 1
X
i0
w
w1 w2 xt d−1
1 −1 e
w 1 x1 t
w2 idt
χx1 e
dx1
e
χi d
w1
X
0 i0
ew1 w2 xt
w1
X
X
χx1 e
w 1 x1 t
dw
1 −1
w2 it
dx1
e χi
X
2.21
i0
1 −1
1 dw
χi χx1 ew1 x1 w2 xw2 /w1 it dx1
w1 i0
X
∞
1 −1
1 dw
w2 w1k tk
χi Bk,χ w2 x i
w1 i0
w1
k!
k0
dw −1
1
tk
w2
k−1
.
χiBk,χ w2 x i w1
w1
k!
i0
k0
∞
From the symmetric property of Tχ w1 , w2 in w1 and w2 , we can also derive the following
equation:
Tχ w1 , w2 ew1 w2 xt
w2
χx2 e
w 2 x2 t
dx2
dw2
X
χx1 ew1 x1 t dx1
edw1 w2 xt dx
dw1 w2 t
w1 w2 xt d−1
e
−1
e
w 2 x2 t
w1 it
χx2 e
dx2
χie
w2
ew1 dt − 1
X
i0
w1 w2 xt w
d−1
2 −1
e
w 2 x2 t
w1 dt
w1 it
χx2 e
dx2
e
χie
w2
X
i0
0
X
X
2 −1
1 dw
χi χx2 ew2 x2 w1 xw1 /w2 it dx2
w2 i0
X
2.22
∞
2 −1
1 dw
w1 w2k tk
χi Bk,χ w1 x i
w2 i0
w2
k!
k0
dw −1
∞
k
2
w1
k−1 t
.
χiBk,χ w1 x i w2
w2
k!
i0
k0
By comparing the coefficients on the both sides of 2.21 and 2.22, we obtain the following
theorem.
Abstract and Applied Analysis
7
Theorem 2.3. For w1 , w2 , d ∈ N, one has
dw
2 −1
w2
w1
k−1
χiBk,χ w2 x i w1 χiBk,χ w1 x i w2k−1 .
w1
w2
i0
dw
1 −1
i0
2.23
Remark 2.4. Let x 0 in Theorem 2.3. Then we see that
dw
1 −1
χiBk,χ
i0
dw
2 −1
w2
w1
i w1k−1 χiBk,χ
i w2k−1 .
w1
w2
i0
2.24
If we take w2 1, then we have
dw
1 −1
χiBk,χ
i0
d−1
i
χiBk,χ w1 i.
w1k−1 w1
i0
2.25
Remark 2.5. Let χ be trivial character. Then we can easily derive the “multiplication theorem
for Bernoulli polynomials” from Theorems 2.2 and 2.3 see 14.
Acknowledgment
The present research has been conducted by the research grant of the Kwangwoon University
in 2009.
References
1 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299,
2002.
2 L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, pp. 987–1000,
1948.
3 M. Cenkci, Y. Simsek, and V. Kurt, “Further remarks on multiple p-adic q-L-function of two variables,”
Advanced Studies in Contemporary Mathematics, vol. 14, no. 1, pp. 49–68, 2007.
4 M. Cenkci, Y. Simsek, and V. Kurt, “Multiple two-variable p-adic q-L-function and its behavior at
s 0,” Russian Journal of Mathematical Physics, vol. 15, no. 4, pp. 447–459, 2008.
5 T. Ernst, “Examples of a q-umbral calculus,” Advanced Studies in Contemporary Mathematics, vol. 16,
no. 1, pp. 1–22, 2008.
6 A. S. Hegazi and M. Mansour, “A note on q-Bernoulli numbers and polynomials,” Journal of Nonlinear
Mathematical Physics, vol. 13, no. 1, pp. 9–18, 2006.
7 T. Kim, “Non-Archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials,”
Russian Journal of Mathematical Physics, vol. 10, no. 1, pp. 91–98, 2003.
8 T. Kim, “Power series and asymptotic series associated with the q-analog of the two-variable p-adic
L-function,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 186–196, 2005.
9 T. Kim, “Multiple p-adic L-function,” Russian Journal of Mathematical Physics, vol. 13, no. 2, pp. 151–
157, 2006.
10 T. Kim, “q-Euler numbers and polynomials associated with p-adic q-integrals,” Journal of Nonlinear
Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007.
11 T. Kim, “A note on p-adic q-integral on Zp associated with q-Euler numbers,” Advanced Studies in
Contemporary Mathematics, vol. 15, no. 2, pp. 133–137, 2007.
12 T. Kim, “q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,”
Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008.
8
Abstract and Applied Analysis
13 T. Kim, “On the symmetry of the q-Bernoulli polynomials,” Abstract and Applied Analysis, vol. 2008,
Article ID 914367, 7 pages, 2008.
14 T. Kim, “Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials,” Journal of
Difference Equations and Applications, vol. 14, no. 12, pp. 1267–1277, 2008.
15 T. Kim, “Note on q-Genocchi numbers and polynomials,” Advanced Studies in Contemporary
Mathematics, vol. 17, no. 1, pp. 9–15, 2008.
16 T. Kim, “Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral
on Zp ,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009.
17 Y.-H. Kim, W. Kim, and L.-C. Jang, “On the q-extension of Apostol-Euler numbers and polynomials,”
Abstract and Applied Analysis, vol. 2008, Article ID 296159, 10 pages, 2008.
18 B. A. Kupershmidt, “Reflection symmetries of q-Bernoulli polynomials,” Journal of Nonlinear
Mathematical Physics, vol. 12, pp. 412–422, 2005.
19 H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, “A note on p-adic q-Euler measure,” Advanced
Studies in Contemporary Mathematics, vol. 14, no. 2, pp. 233–239, 2007.
20 K. H. Park and Y.-H. Kim, “On some arithmetical properties of the Genocchi numbers and
polynomials,” Advances in Difference Equations, vol. 2008, Article ID 195049, 14 pages, 2008.
21 M. Schork, “A representation of the q-fermionic commutation relations and the limit q 1,” Russian
Journal of Mathematical Physics, vol. 12, no. 3, pp. 394–399, 2005.
22 Y. Simsek, “Theorems on twisted L-function and twisted Bernoulli numbers,” Advanced Studies in
Contemporary Mathematics, vol. 11, no. 2, pp. 205–218, 2005.
23 Y. Simsek, “On p-adic twisted q-L-functions related to generalized twisted Bernoulli numbers,”
Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340–348, 2006.
24 Y. Simsek, “Complete sums of h, q-extension of the Eulerpolynomials and numbers,” http://
arxiv.org/abs/0707.2849.
25 Y.-H. Kim and K.-W. Hwang, “Symmetry of power sum and twisted Bernoulli polynomials,”
Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 127–133, 2009.