Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 318639, 6 pages
doi:10.1155/2009/318639
Research Article
Symmetry Properties of Higher-Order
Bernoulli Polynomials
Taekyun Kim,1 Kyung-Won Hwang,2 and Young-Hee Kim1
1
2
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
Department of General Education, Kookmin University, Seoul 136-702, South Korea
Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr and
Kyung-Won Hwang, khwang7@kookmin.ac.kr
Received 11 March 2009; Revised 6 July 2009; Accepted 2 August 2009
Recommended by Patricia J. Y. Wong
We investigate properties of identities and some interesting identities of symmetry for the
Bernoulli polynomials of higher order using the multivariate p-adic invariant integral on Zp .
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 Zp , Qp , and Cp will, respectively,
denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the
completion of algebraic closure of Qp . For x ∈ Cp , we use the notation xq 1 − qx /1 − q.
Let UDZp be the space of uniformly differentiable functions on Zp , and let vp be the
normalized exponential valuation of Cp with |p|p p−vp p 1/p. For q ∈ Cp with |1 − q|p < 1,
the q-Volkenborn integral on Zp is defined as
Iq f 1
fxdμq x lim N N →∞ p
Zp
q
N
−1
p
fxqx ,
f ∈ UD Zp
1.1
x0
see 1, 2. The ordinary p-adic invariant integral on Zp is given by
I1 f lim Iq f q→1
Zp
fxdx
1.2
2
Advances in Difference Equations
see 1–15. Let f 0 dfx/dx|x0 . Then we easily see that
I1 f1 I1 f f 0,
where f1 x fx 1.
1.3
From 1.3, we can derive
Zp
ext dx ∞
tn
t
B
n
et − 1 n0 n!
1.4
see 2, 8–10, where Bn are the nth Bernoulli numbers.
By 1.2 and 1.3, we easily see that
n Zp ext dx 1 xt
xnt
e
dx −
e dx
t
enxt dx
Zp
Zp
Zp
∞
n−1
∞
tk
tk it
k
e i
Sk n − 1 ,
k! k0
k!
n0
i0
i0
n−1
1.5
where Sk n 0k 1k · · · nk for k ∈ Z .
It is known that the Bernoulli polynomials are defined by
Zp
exyt dx ∞
tn
t
xt
,
e
B
x
n
n!
et − 1
n0
1.6
where Bn x are called the nth Bernoulli polynomials. The Bernoulli polynomials of order k,
denoted Bnk x, are defined as
e
xt
t
et − 1
k
t
et − 1
× ··· ×
∞
tn
t
k
xt
e
B
x
n
n!
et − 1
n0
1.7
k
see 3–6. Then the values of Bn x at x 0 are called the Bernoulli numbers of order
k. When k 1, the polynomials or numbers are called the Bernoulli polynomials or
numbers. The purpose of this paper is to investigate some interesting properties of symmetry
for the multivariate p-adic invariant integral on Zp . From the properties of symmetry for
the multivariate p-adic invariant integral on Zp , we derive some interesting identities of
symmetry for the Bernoulli polynomials of higher order.
2. Symmetry Properties of Higher-Order Bernoulli Polynomials
Let w1 , w2 ∈ N. Then we define
D
m
w1 , w2 w1 t
e w1 t − 1
m
e
w1 w2 tx
e
w1 w2 t
−1
w2 t
e w2 t − 1
m
ew1 w2 yt
.
w1 w2 t
2.1
Advances in Difference Equations
3
From 2.1, we note that
Dm w1 , w2 where
Zm
p
Zm
p
ew1 x1 x2 ···xm w2 xt dx1 · · · dxm Zmp ew2 x1 x2 ···xm w1 yt dx1 · · · dxm
,
ew1 w2 xt dx
Zp
fx1 , . . . , xm dx1 · · · dxm In 2.1, we note that D
Dm w1 , w2 m
Zp
···
Zp
fx1 , . . . , xm dx1 · · · dxm .
w1 , w2 is symmetric in w1 , w2 . By 2.1, we see that
⎛
Zm
p
Zp
ew1 x1 ···xm t dx1 · · · dxm ew1 w2 xt ⎝ ×
2.2
ew2 xm t dxm
Zp
ew1 w2 xt dx
⎞
⎠
Zm−1
p
2.3
ew2 x1 ···xm−1 t dx1 · · · dxm−1 ew1 w2 yt .
It is easy to see that
ew1 w2 xt
Zm
p
ew1 x1 ···xm t dx1 · · · dxm w1 t
e w1 t − 1
m
ew1 w2 xt ∞
tn
m
Bn w2 xw1n .
n!
n0
2.4
From 2.1, 2.3, and the above formula, we can derive
D
m
w1 , w2 ∞
0
∞
0
∞
n0
t
m
B w2 xw1
!
t
m
B w2 xw1−1
!
⎛
⎝
∞
wk
Sk w1 − 1 2 tk
k!
k0
∞
w2i i 1
m−1 t
Bi
w1 y
i!
w1
i0
⎞ ⎞
⎛ ⎛ j
m−1 ∞
Bj−k w1 y
tj
j−k
⎝ ⎝ Sk w1 − 1wk w
j!⎠ ⎠
2 2
j!
k! j − k !
j0
k0
⎞
j
j
tn
j n−j−1 m
m−1
Bj−k w1 y ⎠ .
w2 w1
Bn−j w2 x Sk w1 − 1
n!
k
k0
n
n
j0
j
2.5
By the symmetry of Dm w1 , w2 in w1 and w2 , we see that
⎞
⎛ j
∞
n
n
j
tn
j
n−j−1
m
m−1
⎝
w1 w2
Sk w2 − 1Bj−k w2 y ⎠ .
Dm w1 , w2 Bn−j w1 x
n!
j
n0
j0
k0 k
2.6
4
Advances in Difference Equations
By comparing the coefficients on both sides of 2.5 and 2.6, we obtain the following
theorem.
Theorem 2.1. For w1 , w2 ∈ N, n ≥ 0, m ≥ 1, one has
n
n
j
j n−j−1 m
w2 w1
Bn−j w2 x Sk w1
k0
j
j0
− 1
j
k
m−1 Bj−k
w1 y
n
n
j0
j
j
j
j n−j−1 m
m−1 w1 w2
Sk w2 − 1Bj−k w2 y .
Bn−j w1 x
k0 k
2.7
Let y 0 and m 1 in 2.7. Then we have the following corollary.
Corollary 2.2. For n ∈ Z , one has
n
n
j0
j
n−j−1
w1
j
w2 Bn−j w2 xSj w1 − 1
2.8
n
n
j0
j
j n−j−1
w1 w2
Bn−j w1 xSj w2
− 1.
If we take w2 1 in 2.8, then we also obtain the following corollary.
Corollary 2.3. For w1 ∈ N, one has
Bn w1 x n
n
i0
i
w1i−1 Bi xSn−i w1 − 1.
2.9
By the definition of Dm w1 , w2 , we easily see that
m
m−1
w1 w2 t
−1
1
w2 t
w1 t
xw1 w2 t e
e
eyw1 w2 t
w
t
w
t
w
t
1
2
2
w1
e −1
e −1 e −1
w −1
∞
∞
k
1
t
1
w2
m
m−1 kt
B
i w1
B
w2 x w1 y w2
w1 i0 k0 k
w1
k!
!
0
Dm w1 , w2 w −1
k−1
∞
n
1
w1
w2n−k
tn
w2
m
m−1 Bn−k w1 y
n!
w2 x Bk
i
w1
k!
n!
n − k!
n0 k0
i0
∞
n
n
n0
k0
k
m−1 w1k−1 w2n−k Bn−k w1 y
w
1 −1
i0
m
Bk
w2
i
w2 x w1
tn
.
n!
2.10
Advances in Difference Equations
5
From the symmetric property of Dm w1 , w2 in w1 , w2 , we note that
D
m
w1 , w2 ∞
n
n
n0
m−1 w2k−1 w1n−k Bn−k w2 y
k
k0
w
2 −1
i0
m
Bk
n
t
w1
.
i
w1 x w2
n!
2.11
By comparing the coefficients on both sides of 2.10 and 2.11, we obtain the following
theorem.
Theorem 2.4. For w1 , w2 ∈ N, n ∈ Z , m ∈ N, one has
n
n
k0
k
m−1 w1k−1 w2n−k Bn−k
n
n
k
k0
w1 y
1 −1
w
w2
m
Bk
i
w2 x w1
i0
m−1 w2k−1 w1n−k Bn−k w2 y
w
2 −1
i0
m
Bk
w1
i .
w1 x w2
2.12
Let y 0 and m 1 in 2.12. Then we obtain the following Corollary 2.5.
Corollary 2.5. For w1 , w2 ∈ N, one has
w1n−1
w
2 −1
w2
w1
Bn w 2 x i w2n−1
Bn w 1 x i .
w1
w2
i0
w
1 −1
i0
2.13
From 2.12, we can get the well-known result due to Raabe:
1
Bn x i w11−n Bn w1 x.
w1
w
1 −1
i0
2.14
References
1 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.
2 T. Kim, “On a q-analogue of the p-adic log gamma functions and related integrals,” Journal of Number
Theory, vol. 76, no. 2, pp. 320–329, 1999.
3 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards,
1964.
4 Ch. Jordan, Calculus of Finite Differences, Chelsea, New York, NY, USA, 2nd edition, 1950.
5 L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan, London, UK, 1933.
6 N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, Germany, 1924.
7 Y. H. Kim, “On the p-adic interpolation functions of the generalized twisted h, q-Euler numbers,”
International Journal of Mathematical Analysis, vol. 3, pp. 897–904, 2009.
8 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299,
2002.
9 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.
10 T. Kim, “Analytic continuation of multiple q-zeta functions and their values at negative integers,”
Russian Journal of Mathematical Physics, vol. 11, no. 1, pp. 71–76, 2004.
6
Advances in Difference Equations
11 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.
12 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.
13 H. Ozden and Y. Simsek, “A new extension of q-Euler numbers and polynomials related to their
interpolation functions,” Applied Mathematics Letters, vol. 21, no. 9, pp. 934–939, 2008.
14 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.
15 Y.-H. Kim and K.-W. Hwang, “A symmetry of power sum and twisted Bernoulli polynomials,”
Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 127–133, 2009.