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Abstract and Applied Analysis
Volume 2011, Article ID 545314, 8 pages
doi:10.1155/2011/545314
Research Article
A Note on the Modified q-Bernoulli Numbers and
Polynomials with Weight α
T. Kim,1 D. V. Dolgy,2 S. H. Lee,1 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 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 13 July 2011; Accepted 2 October 2011
Academic Editor: Alberto d’Onofrio
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.
A systemic study of some families of the modified q-Bernoulli numbers and polynomials with
weight α is presented by using the p-adic q-integration Zp . The study of these numbers and
polynomials yields an interesting q-analogue related to Bernoulli numbers and polynomials.
1. Introduction
Let p be a fixed prime number. Throughout this paper Zp , Qp , C, and Cp will, respectively,
denote 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 . Let vp be the normalized
exponential valuation of Cp with |p|p p−vp p 1/p. When one talks of q-extension, q is
variously considered as an indeterminate, a complex q ∈ C, or a p-adic number q ∈ Cp . If
q ∈ C, one normally assumes |q| < 1. If q ∈ Cp , then we assume |q − 1|p < p−1/p−1 so that
qx expx log q for |x|p ≤ 1.
The q-number xq is defined by
xq see 1–10.
1 − qx
,
1−q
1.1
2
Abstract and Applied Analysis
We say that f is a uniformly differentiable function at a point a ∈ Zp and denote this
property by f ∈ UDZp , if the difference quotients
fx − f y
Ff x, y x−y
1.2
have a limit l f a as x, y → a, a. c.f. 11.
For f ∈ UDZp , Cp {f | f : Zp → Cp is uniformly differentiable functions}, the
p-adic q-integral on Zp is defined by
Iq f p −1
1 fx dμq x lim N fxqx ,
N
→
∞
p
Zp
q x0
N
1.3
see 3.
From 1.3, we can easily derive the following:
n−1
n−1
q−1 flql ql f l,
qn Iq fn Iq f q − 1
log
q
l0
l0
1.4
where fn x fx n, see 5, 12.
In 1, 2, Carlitz defined a set of numbers Bk,q inductively by
B0,q 1,
⎧
⎨1,
k
qBq 1 − Bk, q ⎩0,
if k 1,
if k > 1,
1.5
with the usual convention about replacing Bqk by Bk,q .
These numbers are the q-extension of ordinary Bernoulli numbers. But they do not remain
finite when q 1. So, Carlitz modified 1.5 as follows:
β0,q 1,
⎧
⎨1,
k
q qβ 1 − βk,q ⎩0,
if k 1,
if k > 1,
1.6
with the usual convention of replacing βk by βk,q .
In 1, Carlitz also considered the extension of Carlitz’s q-Bernoulli numbers as follows:
h
β0,q
h
,
hq
qh
⎧
⎨1,
k
h
qβh 1 − βk,q
⎩0,
if k 1,
if k > 1,
1.7
h
with the usual convention of replacing βh k by βk,q
.
In this paper, we construct the modified q-Bernoulli numbers with weight α, which
are different Carlitz’s q-Bernoulli numbers, by using p-adic q-integral equation. From 1.4,
Abstract and Applied Analysis
3
we derive some interesting identities and relations on the modified q-Bernoulli numbers and
polynomials.
2. The Modified q-Bernoulli Numbers and Polynomials with Weight α
In this section, we assume α ∈ Q. Now, we define the modified q-Bernoulli numbers with
α
weight α Bn,q as follows:
α
Bn,q Zp
q−x xnqα dμq x
1
1 − qα
n
2.1
n
n
l
l0
αl
.
−1l
αlq
Thus, by 2.1, we have
1
α
Bn,q 1−q
∞
q−1
α −n
qαm mn−1
qα .
log q
αq m0
α n
2.2
Therefore, by 2.1 and 2.2, we obtain the following theorem.
Theorem 2.1. For n ∈ Z N ∪ {0}, one has
α
Bn,q
α
1 − qα
n
n
n
l0
l
−1l
l
αlq
∞
q−1
α 1
−
n
qαm mn−1
qα .
n
αq m0
1 − qα log q
2.3
Let us define the generating function of the modified q-Bernoulli numbers with weight
α as follows:
∞
n
α t
Bn,q .
n!
n0
2.4
∞
q − 1 1/1−qα t
α e
−t
qαm emqα t .
log q
αq m0
2.5
α
Fq t Then, by 2.3 and 2.4, we get
α
Fq t 4
Abstract and Applied Analysis
In the viewpoint of 2.1, we define the modified q-Bernoulli numbers with weight α as
follows:
n
α
q−y x y qα dμq y
Bn,q x Zp
n
n
l0
l
α
αlx Bl,q
xn−l
qα q
α n
xqα qαx Bq
,
2.6
for n ∈ Z ,
n
α
α
with the usual convention of replacing Bq by Bn,q .
From 2.6, we note that
α
Bn,q x
α
1−q
α n
n
n
l0
l
−1l qαlx
l
αlq
∞
q−1
α αx −n
q
qαm m xn−1
qα .
n
αq
1 − qα log q
m0
2.7
1
Therefore, by 2.7, we obtain the following theorem.
Theorem 2.2. For n ∈ Z , one has
α
Bn,q x
α
1 − qα
n
n
n
l0
l
−1l qαlx
l
αlq
∞
q−1
α αx −n
q
qαm m xn−1
qα .
n
α
log
q
α
1−q
q
m0
2.8
1
∞ α
α
n
Let Fq t, x n0 Bn,q xt /n! be the generating function of the modified qBernoulli polynomials with weight α.
Then, by 2.7, we get
α
Fq t, x ∞
q − 1 1/1−qα t
α e
−t
qαmx emxqα t .
log q
αq m0
2.9
Therefore, by 2.9, we obtain the following corollary
α
Corollary 2.3. Let Fq t, x α
∞ α
n
n0 Bn,q xt /n!. Then one has
Fq t, x α
∞
q − 1 1/1−qα t
α e
−t
qαmx emxqα t .
log q
αq m0
α
In particular, Fq t, 0 Fq t.
2.10
Abstract and Applied Analysis
5
From Corollary 2.3, we can derive the following equation:
Fqα t, 1 − Fqα t t
α
.
αq
2.11
By 2.5 and 2.11, we get
q−1
α
,
B0,q log q
α
α
Bn,q 1 − Bn,q
⎧ α
⎨
,
αq
⎩
0,
if n 1,
2.12
if n > 1.
Therefore, by 2.12, we obtain the following theorem.
Theorem 2.4. For n ∈ Z , one has
α
B0,q
q−1
,
log q
α
α
Bn,q 1 − Bn,q
⎧ α
⎨
,
αq
⎩
0,
if n 1,
2.13
if n > 1.
By using 2.6, we obtain the following corollary.
Corollary 2.5. For n ∈ Z , one has
α
B0,q
q−1
,
log q
α
qα Bq
⎧ α
⎨
n
,
α
1 − Bn,q αq
⎩
0,
α n
if n 1,
2.14
if n > 1.
α
with the usual convention of replacing Bq by Bn,q .
From 1.4, we can derive the following equation:
Zp
fx nq−x dμq x Zp
fxq−x dμq x n−1
q − 1
f l.
log q l0
2.15
Thus, by 1.6, 2.6, and 2.15, we get
α
α
Bm,q n − Bm,q n−1
α
m lm−1
qαl ,
α
αq l0 q
n ∈ N, m ∈ Z .
2.16
Therefore, by 2.16, we obtain the following theorem.
Theorem 2.6. For n ∈ N, m ∈ Z , one has
α
α
Bm,q n − Bm,q n−1
α
m lm−1
qαl .
α
αq l0 q
2.17
6
Abstract and Applied Analysis
From 2.6, we note that
α
Bn,q x Zp
n
x y qα q−y dμq y
1
lim N N →∞ p
q
N
−1
p
xy
n
qα
y0
p
−1
d−1
n
1−q 1
lim
a x dy qα
1 − qd a0 N → ∞ pN qd y0
N
d−1 dnqα dq
a0
a x
Zp
d
y
n
qαd
2.18
q−dy dμqd y
d−1
x a
dnqα α
.
Bn,qd
d
dq a0
Therefore, by 2.18, we obtain the following distribution relation for the modified qBernoulli polynomials with weight α.
Theorem 2.7. For d ∈ N, n ∈ Z , one has
α
Bn,q x x a
α
.
Bn,qd
d
dq a0
d−1
dnqα 2.19
To derive the relation of reflection symmetry of the modified q-Bernoulli polynomials
with weight α, we evaluate the following p-adic q-integral on Zp :
α
Bn,q−1 1 − x Zp
1 − x x1 nq−α qx1 dμq−1 x1 1
1−
n
q−α
n
n
l0
n
qαn
−1
n
q
1 − qα
l
−1l qαlx−1
n
n
l0
l
αl
αlq
−1l qαlx
2.20
αl
αlq
α
qαn−1 −1n Bn,q x.
Therefore, by 2.20, we obtain the following reflection symmetry relation of the
modified q-Bernoulli polynomials with weight α.
Theorem 2.8. For n ∈ Z , one has
α
α
Bn,q−1 1 − x qαn−1 −1n Bn,q x.
2.21
Abstract and Applied Analysis
7
From 1.3, we note that
1
q
Zp
1 −
xnq−α q−x
n αn−1
dμq x −1 q
Zp
x − 1nqα q−x dμq x
α
−1n qαn−1 Bn,q −1
2.22
α
Bn,q−1 2,
and, by 2.6, we get
n n
α
α
α
Bn,q 2 q2α Bq 2qα qα qα Bq 1 1
n
l
n αl α α
q q Bq 1
l
l0
n
1 l
n αl α α
α
α
α α
q q Bq 1
B0,q nq q Bq 1 l
l2
n
n αl α
q−1
α
α
α
q Bl,q
nq
B1,q log q
αq
l
l2
n
n αl α
α α
q Bl,q .
nq
αq l0 l
2.23
Let n ∈ N with n ≥ 2. Then, by 2.12 and 2.23, we obtain the following theorem.
Theorem 2.9. For n ∈ N with n ≥ 2, one has
n
α
α
α
α
qα Bq 1 Bn,q .
Bn,q 2 − nqα
αq
2.24
In particular,
1
q
n α
α
α
1 − xnq−α q−x dμq x Bn,q−1 2 Bn, q−1 .
q
α
Zp
q
2.25
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.
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1948.
8
Abstract and Applied Analysis
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