Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 948050, 6 pages
doi:10.1155/2012/948050
Research Article
On the Modified q-Bernoulli Numbers of
Higher Order with Weight
T. Kim,1 J. Choi,2 Y.-H. Kim,2 and S.-H. Rim3
1
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea
3
Department of Mathematics Education, Kyungpook National University,
Daegu 702-701, Republic of Korea
2
Correspondence should be addressed to T. Kim, tkkim@kw.ac.kr
Received 11 August 2011; Revised 24 November 2011; Accepted 13 December 2011
Academic Editor: Ferhan M. Atici
Copyright q 2012 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.
The purpose of this paper is to give some properties of the modified q-Bernoulli numbers and
polynomials of higher order with weight. In particular, by using the bosonic p-adic q-integral on
Zp , we derive new identities of q-Bernoulli numbers and polynomials with weight.
1. Introduction
Let p be a fixed odd 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 the algebraic closure of Qp . Let N be the set of natural numbers and Z N ∪ {0}. The p-adic norm of Cp is defined by |p|p 1/p. When one talks of a q-extension, q
can be considered as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp .
Throughout this paper we assume that α ∈ Q and q ∈ Cp with |1 − q|p < p−1/p−1 so that
qx expx log q.
Let UDZp be the space of uniformly differentiable functions on Zp . For f ∈ UDZp ,
the p-adic q-integral on Zp is defined by Kim see 1–3 as follows:
Iq f p −1
1 fxdμq x lim N fxqx ,
N
→
∞
p
Zp
q x0
N
where xq is the q-number of x which is defined by xq 1 − qx /1 − q.
1.1
2
Abstract and Applied Analysis
From 1.1, we have
n−1
n−1
q − 1
ql fl ql f l,
qn Iq fn − Iq f q − 1
log
q
l0
l0
1.2
where fn x fx n see 2–4.
As is well known, Bernoulli numbers are inductively defined by
B0 1,
1
B 1 − Bn 0
n
if n 1,
if n > 1,
1.3
with the usual convention about replacing Bn by Bn see 3, 5.
In 2, 5, 6, the q-Bernoulli numbers are defined by
B0,q
q−1
,
log q
n
qBq 1 − Bn,q 1 if n 1,
0 if n > 1,
1.4
with the usual convention about replacing Bqn by Bn,q . Note that limq → 1 Bn,q Bn . In the
viewpoint of 1.4, we consider the modified q-Bernoulli numbers with weight.
In this paper we study families of the modified q-Bernoulli numbers and polynomials
of higher order with weight. In particular, by using the multivariate p-adic q-integral on Zp ,
we give new identities of the higher-order q-Bernoulli numbers and polynomials with weight.
2. Modified q-Bernoulli Numbers with Weight of Higher Order
For n ∈ Z , let us consider the following modified q-Bernoulli numbers with weight α see
1, 3:
n αl
1
n
,
−1l
n n
αlq
1 − q αq l0 l
Zp
n n
αl
1
n
α
.
Bn,q x x y qα q−y dμq y −1l qαlx
n n
αlq
1 − q αq l0 l
Zp
α
Bn,q xnqα q−x dμq x 2.1
From 2.1, we note that
α
Bn,q x see 1, 3.
n n
αlx α
Bl,q
xn−l
qα q
l
l0
2.2
Abstract and Applied Analysis
3
For k ∈ N and n ∈ Z , by making use of the multivariate p-adic q-integral on Zp , we
k,α
consider the following modified q-Bernoulli numbers with weight α of order k, Bn,q :
k,α
Bn,q 1,α
Zp
···
Zp
α
x1 · · · xk nqα q−x1 −···−xk dμq x1 · · · dμq xk .
k,α
k
2.3
k
Note that Bn,q Bn,q and limq → 1 Bn,q Bn , where Bn are the nth ordinary Bernoulli numbers of order k.
For k, N ∈ N, we have
1−q
1 − qp
N
k pN −1
i1 0
···
N
p
−1
i1 · · · ik nqα
ik 0
k N
n p
−1 n n
−1j qαi1 ···ik j
N
j
1 − qp
i1 ,...,ik 0 j0
k N
N
n
n
1−q
1 − qαp j
1 − qαp j
1
j
···
.
−1 n
N k
1 − qαj
1 − qαj
1 − q αnq j0 j
1 − qp
1−q
1
1 − qα
2.4
k−times
By 1.1, 2.3, and 2.4, we get
k
n 1
n
j αj
−1 k .
n
1 − q αnq j0 j
αj q
k,α
Bn,q 2.5
Therefore, by 2.5, we obtain the following theorem.
Theorem 2.1. For n ≥ 0, one has
k
n 1
n
j αj
−1 k .
n
1 − q αnq j0 j
αj q
k,α
Bn,q 2.6
Let us consider the modified q-Bernoulli and polynomials with weight α of order k as
follows:
k,α
···
2.7
Bn,q x x x1 · · · xk nqα q−x1 −···−xk dμq x1 · · · dμq xk .
Zp
Zp
By the same method of 2.5, we obtain the following theorem.
Theorem 2.2. For n ∈ Z , one has
k
n 1
n
j αxj αj
−1 q k .
n
1 − q αnq j0 j
αj
k,α
Bn,q x q
2.8
4
Abstract and Applied Analysis
By Theorem 2.2, we get
k
n n
j αj
−1 k q−αjk−x
n
1 − q−α j0 j
αj q−1
k
n q−1 q − 1 αj
−1n qαn n
j
q−αjk−x
−1 αj
n
q − 1 q−αj
1 − qα j0 j
k
n −1n qαn n
j αjx −k αj
−1 q q k
n
1 − qα j0 j
αj q
1
k,α
Bn,q−1 k − x 2.9
k,α
−1n qαn−k Bn,q x.
Therefore, by 2.9, we obtain the following theorem.
Theorem 2.3. For n ∈ Z , one has
k,α
k,α
Bn,q−1 k − x −1n qαn−k Bn,q x,
k,α
k,α
Bn,q−1 k −1n qαn−k Bn,q .
2.10
From Theorem 2.3, we note that
k,α
lim B −1 k
q → 1 n,q
k
k
− x Bn k − x,
k
k,α
lim B −1 k
q → 1 n,q
k
−1n Bn .
2.11
k
Thus, we have Bn k −1n Bn , where Bn are the nth Bernoulli numbers of order k.
From 2.3 and 2.7, we can derive the following equations:
−1
m−1
p
1
x i1 · · · il mn1 · · · nl kqα
l N l
N →∞
mq p qm i1 ,...,il 0 n1 ,...,nl 0
k
mkqα m−1
x i1 · · · il
x
·
·
·
·
·
·
x
1
l
m
mlq i1 ,...,il 0 Zp
qαm
Zp
N
l,α
Bk,q x lim
×q
−mx1 −···−mxl
l,αm x i1 · · · il .
B
k,q
m
mlq i1 ,...,il 0
2.12
dμqm x1 · · · dμqm xk mkqα m−1
Therefore, by 2.12, we obtain the following theorem.
Theorem 2.4. For k ∈ Z and l, m ∈ N, one has
l,α
Bk,q x l,α x i1 · · · il
Bk,qm
.
m
mlq i1 ,...,il 0
mkqα m−1
2.13
Abstract and Applied Analysis
5
In particular,
l,α
Bk,q mx l,αm x i1 · · · il .
B
k,q
m
mlq i1 ,...,il 0
mkqα m−1
2.14
From 1.2, we can derive the following integral:
−x
Zp
Zp
fx 1q dμq x fx 2q−x dμq x Zp
Zp
fxq−x dμq x q−1 f 0,
log q
f1 xq−x dμq x q−1 f 1
log q
fxq−x dμq x Zp
2.15
q − 1 f 0 f 1 .
log q
Continuing this process, we obtain
Zp
fx nq−x dμq x Zp
n−1
q − 1
f l.
log q l0
2.16
n−1
mα qαl .
lm−1
α
αq l0 q
2.17
fxq−x dμq x By 2.16, we get
Zp
−x
x nm
qα q dμq x Zp
−x
xm
qα q dμq x Therefore, by 2.1 and 2.17, we obtain the following theorem.
Theorem 2.5. For n ∈ N and m ∈ Z , one has
n−1
α α
α
Bm,q n − Bm,q m
lmα qαl .
αq l0 q
2.18
In an analogues manner as the previous investigation 7–10, we can define a further
generalization of modified q-Bernoulli numbers with weight. Let χ be the Dirichlet character
with conductor d ∈ N. Then the generalized q-Bernoulli numbers with weight attached to χ
can be defined as follows:
α
Bn,χ,q X
χxxnqα q−x dμq x
d−1
a
dnqα α
.
χaBn,qd
d
dq a0
2.19
α
We expect to investigate these objects in future papers. This definition Bn,q was also given in
a previous paper see 9.
6
Abstract and Applied Analysis
Acknowledgments
The authors express their sincere gratitude to the referees for their valuable suggestions and
comments. This paper is supported in part by the Research Grant of Kwangwoon University
in 2011.
References
1 S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionic p-adic q-integral representation on Zp
associated with weighted q-Bernstein and q-Genocchi polynomials,” Abstract and Applied Analysis,
vol. 2011, Article ID 649248, 7 pages, 2011.
2 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299,
2002.
3 T. Kim, “On the weighted q-Bernoulli numbers and polynomials,” Advanced Studies in Contemporary
Mathematics, vol. 21, no. 2, pp. 207–215, 2011.
4 C. S. Ryoo and Y. H. Kim, “A numerical investigation on the structure of the roots of the twisted
q-Euler polynomials,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 131–141, 2009.
5 L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, pp. 987–1000,
1948.
6 Y. Simsek, “Special functions related to Dedekind-type DC-sums and their applications,” Russian
Journal of Mathematical Physics, vol. 17, no. 4, pp. 495–508, 2010.
7 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.
8 T. Kim, D. V. Dolgy, S. H. Lee, B. Lee, and S. H. Rim, “A note on the modified q-Bernoulli numbers
and polynomials with weight α,” Abstract and Applied Analysis, vol. 2011, Article ID 545314, 8 pages,
2011.
9 T. Kim, D. V. Dolgy, B. Lee, and S.-H. Rim, “Identities on the Weighted q-Bernoulli numbers of higher
order,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 918364, 6 pages, 2011.
10 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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