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Abstract and Applied Analysis
Volume 2011, Article ID 361484, 8 pages
doi:10.1155/2011/361484
Research Article
Some Identities on the q-Bernoulli Numbers and
Polynomials with Weight 0
T. Kim, J. Choi, and Y. H. Kim
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Correspondence should be addressed to J. Choi, jeschoi@kw.ac.kr
Received 16 August 2011; Accepted 27 September 2011
Academic Editor: Josef Diblı́k
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.
Recently, Kim 2011 has introduced the q-Bernoulli numbers with weight α. In this paper, we
consider the q-Bernoulli numbers and polynomials with weight α 0 and give p-adic q-integral
representation of Bernstein polynomials associated with q-Bernoulli numbers and polynomials
with weight 0. From these integral representation on Zp , we derive some interesting identities on
the q-Bernoulli numbers and polynomials with weight 0.
1. Introduction
Let p be a fixed prime number. Throughout this paper, Zp , Qp , and Cp will denote the ring
of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic
closure of Qp , respectively. Let N be the set of natural numbers and Z N ∪ {0}.
Let | · |p be a p-adic norm with |x|p p−r , where x pr s/t and p, s p, t s, t 1,
r ∈ Q. In this paper, we assume that q ∈ Cp with |1 − q|p < p−1/p−1 so that qx expx log q,
and xq 1 − qx /1 − 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 as follows:
N
−1
p
Iq f fxdμq x lim
fxμq x pN Zp
N →∞
Zp
1
lim N
N →∞ p
N
−1
p
x0
1.1
x
fxq ,
q x0
see 1–5. For n ∈ N, let fn x fx n. From 1.1, we note that
n−1
n−1
q − 1
ql fl ql f l,
qn Iq fn − Iq f q − 1
log
q
l0
l0
1.2
2
Abstract and Applied Analysis
where f l dfx/dx|xl , see 3, 6, 7. In the special case, n 1, we get
q
Zp
fx 1dμq x −
q−1 f 0.
fxdμq x q − 1 f0 log
q
Zp
1.3
Throughout this paper, we assume that α ∈ Q.
The q-Bernoulli numbers with weight α are defined by Kim 8 as follows:
⎧ α
⎪
⎨
n
α
α
q qα βq 1 − βn,q αq
⎪
⎩0
α
β0,q 1,
α
if n 1,
1.4
if n > 1,
α
with the usual convention about replacing βq n with βn,q . From 1.4, we can derive the
following equation:
α
βn,q
n
n
αl 1
1
−1l
n n
αl 1q
1 − q αq l0 l
∞
∞
nα −
qmαm mn−1
qm mnqα .
qα 1 − q
αq m0
m0
1.5
By 1.1, 1.4, and 1.5, we get
α
βn,q
xnqα dμq x
Zp
n
n
αl 1
1
.
−1l
n n
αl 1q
1 − q αq l0 l
1.6
The q-Bernoulli polynomials with weight α are defined by
α
βn,q x
Zp
xy
n
dμq y
qα
n
n
l0
l
α
qαlx xn−l
qα βl,q .
1.7
By 1.6 and 1.7, we easily see that
α
βn,q x
n
n
αl 1
1
.
−1l qαlx
n n
αl 1q
1 − q αq l0 l
1.8
Let CZp be the set of continuous functions on Zp . For f ∈ CZp , the p-adic analogue
of Bernstein operator of order n for f is given by
Bn,q
n n n
k
k
xk 1 − xn−k ,
f
f
f |x Bk,n x n
n
k
k0
k0
1.9
Abstract and Applied Analysis
3
where n, k ∈ Z see 1, 9, 10. For n, k ∈ Z , the p-adic Bernstein polynomials of degree n
are defined by Bk,n x nk xk 1 − xn−k for x ∈ Zp , see 1, 10, 11.
In this paper, we consider Bernstein polynomials to express the p-adic q-integral on
Zp and investigate some interesting identities of Bernstein polynomials associated with the
q-Bernoulli numbers and polynomials with weight 0 by using the expression of p-adic qintegral on Zp of these polynomials.
2. q-Bernoulli Numbers with Weight 0 and Bernstein Polynomials
0
In the special case, α 0, the q-Bernoulli numbers with weight 0 will be denoted by βn,q βn,q .
From 1.4, 1.5, and 1.6, we note that
∞
βn,q
n0
∞
tn n! n0
Zp
q−1
log q
xn dμq x
tn
n!
Zp
ext dμq x
2.1
t log q
.
qet − 1
It is easy to show that
t log q
t
t
q−1
qe − 1
1 − q−1
et − q−1
log q
q−1
1 − q−1
et − q−1
∞
∞
tn log q tn
t Hn q−1
Hn q−1
q − 1 n0
n! q − 1 n0
n!
∞
∞
tn log q tn
1 ,
nHn−1 q−1
Hn q−1
q − 1 n1
n! q − 1 n0
n!
2.2
where Hn q−1 are the nth Frobenius-Euler numbers.
By 2.1 and 2.2, we get
βn,q ⎧
⎪
⎨1
if n 0,
n
⎪
⎩
Hn−1 q
log q
−1
Hn q
−1
if n > 0.
2.3
Therefore, we obtain the following theorem.
Theorem 2.1. For n ∈ Z , we have
βn,q ⎧
⎪
⎨1
n
⎪
⎩
Hn−1 q−1 Hn q−1
log q
where Hn q−1 are the nth Frobenius-Euler numbers.
if n 0,
if n > 0,
2.4
4
Abstract and Applied Analysis
From 1.5, 1.6, and 1.7, we have
⎧
q−1
⎪
⎨
n
q βq 1 − βn,q log q
⎪
⎩0
β0,q 1,
if n 1,
2.5
if n > 1,
with the usual convention about replacing βq n with βn,q . By 1.7, the nth q-Bernoulli
polynomials with weight 0 are given by
βn,q x n
n
xn−l βl,q .
x y dμq y l
l0
Zp
n
2.6
From 2.6, we can derive the following function equation:
q−1
log q
∞
t log q xt tn
.
e
β
x
n,q
n!
qet − 1
n0
2.7
Thus, by 2.7, we get that
for n ∈ Z .
2.8
x − 1n dμq x −1n βn,q −1.
2.9
βn,q−1 1 − x −1n βn,q x,
By the definition of p-adic q-integral on Zp , we see that
Zp
1 − xn dμq x −1n
Zp
Therefore, by 2.8 and 2.9, we obtain the following theorem.
Theorem 2.2. For n ∈ Z , we have
2.10
−1n βn,q x βn,q−1 1 − x.
In particular, x −1, we get
Zp
n
1 − y dμq y −1n βn,q −1 βn,q−1 2.
2.11
From 2.5, we can derive the following equation:
q2 βn,q 2 q2 nq
q−1
− q βn,q ,
log q
Therefore, by 2.12, we obtain the following theorem.
if n > 1.
2.12
Abstract and Applied Analysis
5
Theorem 2.3. For n ∈ N with n > 1, we have
βn,q 2 1 1
1
n q−1
− 2 βn,q .
q log q
q q
2.13
Taking the p-adic q-integral on Zp for one Bernstein polynomials in 1.9, we get
Zp
Bk,n xdμq x n
k
xk 1 − xn−k dμq x
Zp
n−k n − k
n k
−1l
l
l0
Zp
2.14
xkl dμq x
n−k n − k
n k
l
l0
−1l βkl,q .
From the symmetry of Bernstein polynomials, we note that
Zp
Bk,n xdμq x Zp
Bn−k,n 1 − xdμq x
k
n k
k
l
l0
2.15
−1kl 1 − xn−l dμq x.
Zp
Let n > k 1. Then, by Theorem 2.3 and 2.15, we get
Zp
Bk,n xdμq x k
k
n k
l0
l
−1
kl
n−l
1 − −1
q
q−1 − 1
log q
⎧
q−1
⎪
⎪
⎪
− q q2 βn,q−1
1
n
⎪
⎪
log
q
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 1−q
nq2 βn,q−1 nq2 βn−1,q−1
n
log
q
⎪
⎪
⎪
⎛ ⎞
⎛ ⎞
⎪
⎪
⎪
k
⎪
n
k
⎪
⎪
⎪⎝ ⎠q2 ⎝ ⎠−1kl βn−l,q−1
⎪
⎪
⎩ k
l
l0
− q q βn−l,q−1
2
if k 0,
2.16
if k 1,
if k > 1.
By comparing the coefficients on the both sides of 2.14 and 2.16, we obtain the
following theorem.
6
Abstract and Applied Analysis
Theorem 2.4. For n, k ∈ Z with n > k 1, we have
n−1 n − 1
−1l β1l,q l
l0
n−k n − k
l0
l
−1l βkl,q q
1−q
q2 βn,q−1 q2 βn−1,q−1 ,
log q
2.17
k
k
2
l
l0
−1kl βn−l,q−1 ,
if k > 1.
In particular, when k 0, we have
n
n
−1l βl,q 1 n
l
l0
q−1
− q q2 βn,q−1 .
log q
2.18
Let m, n, k ∈ Z with m n > 2k 1. Then we see that
Zp
Bk,n xBk,m xdμq x
n
m
k
k
Zp
x2k 1 − xnm−2k dμq x
2k
2k
n
m k
k
l0
−1l2k
l
2k
2k
n
m k
k
l0
l
Zp
−1l2k
1 − xnm−l dμq x
1−q
1 − n m − l
log q
⎧
q−1
⎪
⎪
− q q2 βnm,q−1
1
m
⎪
n
⎪
⎪
log q
⎪
⎨
⎛ ⎞
⎛ ⎞⎛ ⎞
2k
⎪
n
m
2k
⎪
⎪
2
⎝ ⎠−1l2k βnm−l,q−1
⎪⎝ ⎠⎝ ⎠q
⎪
⎪
⎩ k
k
l
l0
2.19
− q q2 βnm−l,q−1
if k 0,
if k > 0.
For m, n, k ∈ Z , we have
Zp
Bk,n xBk,m xdμq x n
m
k
k
Zp
x2k 1 − xnm−2k dμq x
n m − 2k
n
m nm−2k
k
k
l0
l
−1l
n m − 2k
n
m nm−2k
k
k
l0
l
Zp
x2kl dμq x
−1l βl2k,q .
2.20
Abstract and Applied Analysis
7
By comparing the coefficients on the both sides of 2.19 and 2.20, we obtain the
following theorem.
Theorem 2.5. For m, n, k ∈ Z with m n > 2k 1, we have
nm
nm
l0
l
−1l βl,q 1 n m
q−1
log q
− q q2 βnm,q−1 .
2.21
−1l2k βnm−l,q−1 .
2.22
In particular, when k / 0, we have
nm−2k
n m − 2k
l
−1 βl2k,q q
l
l0
2
2k
2k
l0
l
For s ∈ N, let k, n1 , . . . , ns ∈ Z with n1 n2 · · · ns > sk 1. By the same method
above, we get
⎧
s
⎪
q−1
⎪
⎪
− q q2 βn1 n2 ···ns ,q−1
n
1
⎪
i
⎪
⎪
log
q
⎪
i1
s
⎨
Bk,ni x dμq x ⎛ ⎛ ⎞⎞
⎛ ⎞
⎪
s
sk
⎪
Zp
ni
sk
i1
⎪
⎪
⎝ ⎝ ⎠⎠q2 ⎝ ⎠−1lsk βn n ···n −l,q−1
⎪
⎪
1
2
s
⎪
⎩ i1 k
l
l0
if k 0,
if k > 0.
2.23
From the binomial theorem, we note that
s
Zp
Bk,ni x dμq x i1
n ···n −sk
s
1 s
ni
n1 · · · ns − sk
−1l βlsk,q .
l
k
i1
l0
2.24
By comparing the coefficients on the both sides of 2.23 and 2.24, we obtain the
following theorem.
Theorem 2.6. For s ∈ N, let k, n1 , . . . , ns ∈ Z with n1 n2 · · · ns > sk 1. Then, we have
n1 ···n
s
n1 · · · ns
l
l0
l
−1 βl,q
s
q−1
1
ni
− q q2 βn1 ···ns ,q−1 .
log q
i1
2.25
In particular, when k /
0, we have
n1 ···n
s −sk
l0
n1 · · · ns − sk
l
l
−1 βlsk,q q
2
sk
sk
l0
l
−1lsk βn1 ···ns −l,q−1 .
2.26
8
Abstract and Applied Analysis
Acknowledgment
The present research has been conducted by the Research Grant of Kwangwoon University
in 2011.
References
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Mathematics, vol. 21, pp. 207–215, 2011.
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