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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 918364, 6 pages
doi:10.1155/2011/918364
Research Article
Identities on the Weighted q-Bernoulli Numbers of
Higher Order
T. Kim,1 D. V. Dolgy,2 B. Lee,3 and S.-H. Rim4
1
Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea
Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea
3
Department of Wireless of Communications Engineering, Kwangwoon University,
Seoul 139-701, Republic of Korea
4
Department of Mathematics Education, Kyungpook National University,
Taegu 702-701, Republic of Korea
2
Correspondence should be addressed to T. Kim, tkkim@kw.ac.kr
Received 6 August 2011; Accepted 17 August 2011
Academic Editor: Lee-Chae Jang
Copyright q 2011 T. 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 give a new construction of the weighted q-Bernoulli numbers and polynomials of higher order
by using multivariate p-adic q-integral on Zp .
1. Introduction
Let p be a fixed prime number. Throughout this paper, Z, Zp , Qp , C, and Cp will, respectively,
denote the ring of rational integers, the ring of p-adic rational integers, the field of p-adic
rational numbers, the complex number field, and the completion of the algebraic closure of
Qp . The p-adic norm of Cp is defined by |x|p 1/pr where x pr s/t with p, s p, t s, t 1, r ∈ Q. In this paper, we assume α ∈ Q and Z N ∪ {0}. Let q ∈ Cp with |q − 1|p <
p−1/p−1 and let xq 1 − qx /1 − q. Note that limq → 1 xq x see 1–13. Recently, the
q-Bernoulli numbers with weight α are defined by
α
β0,q
1,
⎧ α
⎨
n
α
q qα βqα 1 − βn,q αq
⎩
0
if n 1,
if n > 1,
α
with the usual convention about replacing βα n by βn,q , see 4.
1.1
2
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The q-Bernoulli polynomials with weight α are also defined by as
α
βn,q x n
n
αlx βl,q
xn−l
qα q
α
l
l0
xqα qαx βq
α
n
1.2
for n ≥ 0.
,
For f ∈ UDZp the space of uniformly differentiable functions on Zp , the p-adic q-integral
on Zp is defined by
Iq f 1
Zp
fxdμq x lim N N →∞ p
q
N
−1
p
fxqx ,
1.3
x0
see 4–12. From 1.3, we note that
qn Iq fn Iq f q − 1
n−1
ql fl l0
n−1
q − 1
ql f l,
log q l0
1.4
where fn x fx n, see 1–12.
We have the Witt formula for the q-Bernoulli numbers and polynomials with weight α
as follows see 4, 5, 12:
α
βn,q Zp
α
βn,q x xnqα dμq x,
Zp
n
x y qα dμq x.
1.5
αl 1
,
αl 1q
1.6
From 1.4 and 1.5, we have
α
βn,q x n
1
1−q
n
αnq l0
−1l qαlx
see 4. By 1.6, we easily get limq → 1 βn,q x Bn x, where Bn x are the Bernoulli polynomials of degree n.
To give the new construction of the weighted q-Bernoulli numbers and polynomials of
higher order, we first use the multivariate p-adic q-integral on Zp . The purpose of this paper
is to give the higher-order q-Bernoulli numbers and polynomials with weight α and to derive
a new explicit formulas by these numbers and polynomials.
α
Discrete Dynamics in Nature and Society
3
2. On the Higher Order q-Bernoulli Numbers with Weight α
For hi i 1, 2, . . . , k ∈ Z , we consider a sequence of p-adic rational numbers as expansion
of the weighted q-Bernoulli numbers and polynomials of order k as follows:
k,α
βn,q h1 , h2 , . . . , hk k,α
βn,q h1 , h2 , . . . , hk | x Zp
Zp
···
Zp
···
Zp
x1 · · · xk nqα q
k
j1
xj hj −1
x x1 · · · xk nqα q
k
j1
dμq x1 · · · dμq xk ,
xj hj −1
2.1
dμq x1 · · · dμq xk .
2.2
From 2.1 and 2.2, we can derive the following equations:
k,α
βn,q h1 , h2 , . . . , hk n
n
1
1−q
n
αnq l0
l
−1l
αl h1 αl h2 · · · αl hk , 2.3
αl h1 q αl h2 q · · · αl hk q
k,α
βn,q h1 , h2 , . . . , hk | x
n
n
1
n
1 − q αnq
n
n
l0
l
l
l0
−1l qαlx
αl h1 αl h2 · · · αl hk ,
αl h1 q αl h2 q · · · αl hk q
αlx βl,q h1 , h2 , . . . , hk .
xn−l
qα q
k,α
2.4
By 2.3 and 2.4, we get
n
n
l0
l
−1l qαlx
n
n
l0
l
αl h1 αl h2 · · · αl hk ,
αl h1 q αl h2 q · · · αl hk q
αlx
1−q
xn−l
qα q
n−l
l
l
n−l
αq
s0
2.5
αs h1 · · · αs hk .
−1s
h1 q · · · αs hk q
αs
s
Therefore, by 2.5, we obtain the following theorem.
Theorem 2.1. For hi i 1, 2, . . . , k ∈ Z , and k ∈ N, we have
n
n
l0
l
−1l qαlx
n
n
l0
l
αl h1 αl h2 · · · αl hk αl h1 q αl h2 q · · · αl hk q
αlx
1−q
xn−l
qα q
n−l
l
l
n−l
αq
s0
αs h1 · · · αs hk .
−1s
αs h1 q · · · αs hk q
s
2.6
4
Discrete Dynamics in Nature and Society
From 1.3 and 1.4, we note that
qh1
x1 x 1nqα qx1 h1 −1 dμq x1 Zp
Zp
x1 xnqα qx1 h1 −1 dμq x1 xnqα h1
q−1 nxn−1
qα
α
.
αq
2.7
Therefore, by 2.7, we obtain the following theorem.
Theorem 2.2. For n ∈ Z , we have
qh1 βn,q h1 | x 1 − βn,q h1 | x h1 q − 1 xnqα nxn−1
qα
1,α
1,α
α
.
αq
2.8
From 2.2 and 2.3, we have
qαx βn,q αh1 α, αh2 α, . . . , αhk α | x
k,α
qαx
···
Zp
qα − 1
Zp
Zp
Zp
···
Zp
x1 · · · xk xnqα q
···
Zp
k
j1
xj αhj α−1
x1 · · · xk xn1
qα q
x1 · · · xk xnqα q
k
j1
k
xj αhj −1
j1
xj αhj −1
dμq x1 · · · dμq xk dμq x1 · · · dμq xk 2.9
dμq x1 · · · dμq xk .
Therefore, by 2.9, we obtain the following proposition.
Proposition 2.3. For n ∈ Z , we have
qαx βn,q αh1 α, αh2 α, . . . , αhk α | x
k,α
k,α
k,α
qα − 1 βn1,q αh1 , . . . , αhk | x βn,q αh1 , . . . , αhk | x.
By 2.2, we get
k,α
βn,q h1 , h2 , . . . , hk | x
Zp
···
Zp
x x1 · · · xk nqα q
k
j1
xj hj −1
dμq x1 · · · dμq xk 2.10
Discrete Dynamics in Nature and Society
dnqα
d−1
dkq
s1 ,...,sk 0
q
k
j1
hj sj
Zp
5
s · · · s x
n
1
k
x1 · · · xk
d
qdα
Zp
···
× qd
dnqα
d−1
dkq
s1 ,...,sk 0
q
k
j1
hj sj k,α
βn,qd
k
j1
h1 , . . . , hk |
xj hj −1
dμqd x1 · · · dμqd xk s1 · · · sk x ,
d
2.11
where d ∈ N.
Therefore, by 2.11, we obtain the following theorem.
Theorem 2.4. For n ∈ Z and d ∈ N, we have
k,α
βn,q h1 , h2 , . . . , hk | x dnqα
d−1
dkq
s1 ,...,sk 0
q
k
j1
hj sj k,α
βn,qd
h1 , . . . , hk |
s1 · · · sk x .
d
2.12
Let d> 0 be a fixed integer. For N ∈ N, we set
X Xd lim
←−
N
X∗ Z
,
dpN Z
X1 Zp ,
a dpZp ,
2.13
0<a<dp
a,p1
a dpN Zp x ∈ X | x ≡ a mod dpN ,
where a ∈ Z satisfies the condition 0 ≤ a < dpN .
Let χ be a primitive Dirichlet character with conductor d ∈ N. Then we consider the
generalized q-Bernoulli numbers with weight α of order k as follows:
k,α
βn,χ,q h1 , h2 , . . . , hk ···
X
k
k
χxi x1 · · · xk nqα q i1 xi hi −1 dμq x1 · · · dμq xk .
X i1
2.14
From 2.14, we have
⎛
⎞
k
s1 · · · sk k,α
j1 hj sj ⎝
⎠βk,α
h
.
q
χ
s
,
.
.
.
,
h
|
βn,χ,q h1 , h2 , . . . , hk j
1
k
d
n,q
d
dkq s1 ,...,sk 0
j1
dnqα
d−1
k
2.15
6
Discrete Dynamics in Nature and Society
References
1 L. Carlitz, “Expansions of q-Bernoulli numbers,” Duke Mathematical Journal, vol. 25, pp. 355–364, 1958.
2 A. Bayad and T. Kim, “Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials,”
Russian Journal of Mathematical Physics, vol. 18, pp. 133–143, 2011.
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Mathematics, vol. 21, pp. 207–215, 2011.
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2002.
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q-Euler numbers,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 1, pp. 93–100, 2009.
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numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.
10 C. S. Ryoo, “On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots
of unity,” Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 2, pp. 255–263, 2010.
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Journal of Mathematical Physics, vol. 17, no. 4, pp. 495–508, 2010.
12 Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with
their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–
278, 2008.
13 H. M. Srivastava, T. Kim, and Y. Simsek, “q-Bernoulli numbers and polynomials associated with
multiple q-zeta functions and basic L-series,” Russian Journal of Mathematical Physics, vol. 12, no. 2,
pp. 241–268, 2005.
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