Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 513821, 8 pages
doi:10.1155/2011/513821
Research Article
A Study on the p-Adic q-Integral Representation on
p Associated with the Weighted q-Bernstein and
q-Bernoulli Polynomials
T. Kim,1 A. Bayad,2 and Y.-H. Kim1
1
2
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Département de Mathématiques, Université d’Evry Val d’Essonne, Boulevard François Mitterrand,
91025 Evry Cedex, France
Correspondence should be addressed to A. Bayad, abayad@maths.univ-evry.fr
Received 6 December 2010; Accepted 15 January 2011
Academic Editor: Vijay Gupta
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 investigate some interesting properties of the weighted q-Bernstein polynomials related to the
weighted q-Bernoulli numbers and polynomials by using p-adic q-integral on p.
1. Introduction and Preliminaries
Let p be a fixed prime number. Throughout this paper, p, p , and p will denote the ring
of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic
closure of p , respectively. Let be the set of natural numbers, and let ∪ {0}. Let νp
be the normalized exponential valuation of p with |p|p p−νp p 1/p. Let q be regarded
as either a complex number q ∈ or a p-adic number q ∈ p . If q ∈ , then we always
assume |q| < 1. If q ∈ p , we assume that |1 − q|p < 1. In this paper, we define the q-number
as xq 1 − qx /1 − q see 1–13.
Let C0, 1 be the set of continuous functions on 0, 1. For α ∈ and n, k ∈ , the
weighted q-Bernstein operator of order n for f ∈ C0, 1 is defined by
α
α n,q f
n n
n k
k
α Bk,n x, q .
f
f
|x xkqα 1 − xn−k
−α
q
n
n
k
k0
k0
1.1
Here Bk,n x, q is called the weighted q-Bernstein polynomials of degree n see 2, 5, 6.
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Journal of Inequalities and Applications
Let UD p be the space of uniformly differentiable functions on p. For f ∈ UD
the p-adic q-integral on p, which is called the bosonic q-integral on p, is defined by
Iq f p −1
1 fxdμq x lim N fxqx ,
N →∞ p
p
q x0
p,
N
1.2
see 10.
The Carlitz’s q-Bernoulli numbers are defined by
β0,q 1,
q qβ 1
k
− βk,q 1,
if k 1,
0,
if k > 1,
1.3
with the usual convention about replacing βk by βk,q see 3, 9, 10. In 3, Carlitz also defined
the expansion of Carlitz’s q-Bernoulli numbers as follows:
h
β0,q
h
,
hq
q
h
qβ 1
h
n
−
h
βn,q
1, if n 1,
0, if n > 1,
1.4
n
h
.
with the usual convention about replacing βh by βn,q
The weighted q-Bernoulli numbers are constructed in previous paper 6 as follows:
for α ∈ ,
α
β0,q 1,
q qα βα 1
n
α
− βn,q
⎧ α
⎨
,
αq
⎩
0,
if n 1,
1.5
if n > 1,
n
α
with the usual convention about replacing βα by βn,q . Let fn x fx n. By the
definition 1.2 of p-adic q-integral on p, we easily get
p −1
1 qIq f1 q lim N fx 1qx ,
N→∞ p
q x0
N
N
pN −1
f pN qp − f0
1 lim N fxqx lim
N
N →∞ p
N →∞
p q
q x0
q−1 f 0,
fxdμq x q − 1 f0 log q
p
1.6
Continuing this process, we obtain easily the relation
q
fn xdμq x −
n
p
n−1
n−1
q − 1
fxdμq x q − 1
ql fl ql f l,
log q l0
p
l0
where n ∈ and f l dfl/dx see 6.
1.7
Journal of Inequalities and Applications
3
Then by 1.2, applying to the function x → xnqα , we can see that
α
βn,q xnqα dμq x −
p
∞
∞
nα qmαm mn−1
qm mnqα .
qα 1 − q
αq m0
m0
1.8
The weighted q-Bernoulli polynomials are also defined by the generating function as
follows:
α
Fq t, x −t
∞
∞
∞
α tn
α
βn,q x ,
qmαm emxqα t 1 − q
qm emxqα t n!
αq m0
m0
n0
1.9
see6. Thus, we note that
α
βn,q x
n
n
αlx α
βl,q
xn−l
qα q
l
l0
∞
∞
nα −
qmαm m xn−1
qm m xnqα .
qα 1 − q
αq m0
m0
1.10
From 1.2 and the previous equalities, we obtain the Witt’s formula for the weighted
q-Bernoulli polynomials as follows:
α
βn,q x p
n
n αlx n−l
x y qα dμq y q xqα
l
l0
n
l
y qα dμq y .
1.11
p
By using 1.2 and the weighted q-Bernoulli polynomials, we easily get
n−1
n−1
mα α
α
qn βm,q n − βm,q q − 1
ql lm
qαll lm−1
qα qα ,
α
q l0
l0
1.12
where n, α ∈ and m ∈ see 6.
In this paper, we consider the weighted q-Bernstein polynomials to express the bosonic
q-integral on p and investigate some properties of the weighted q-Bernstein polynomials
associated with the weighted q-Bernoulli polynomials by using the expression of p-adic qintegral on p of those polynomials.
2. Weighted q-Bernstein Polynomials and q-Bernoulli Polynomials
In this section, we assume that α ∈ and q ∈ p with |1 − q|p < 1.
Now we consider the p-adic weighted q-Bernstein operator as follows:
α n,q f
n n
n k
k
α k
n−k
| x fx Bk,n x, q .
f
f
xqα 1 − xq−α n
n
k
k0
k0
2.1
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Journal of Inequalities and Applications
The p-adic q-Bernstein polynomials with weight α of degree n are given by
α Bk,n x, q
n
k
xkqα 1 − xn−k
q−α ,
2.2
where x ∈ p, α ∈ , and n, k ∈ see 6, 7. Note that Bk,n x, q Bn−k,n 1 − x, 1/q. That
is, the weighted q-Bernstein polynomials are symmetric.
From the definition of the weighted q-Bernoulli polynomials, we have
α
α
α
α
βn,q−1 1 − x −1n qαn βn,q x.
p,
By the definition of p-adic q-integral on
2.3
we get
1 − xnq−α dμq x qαn −1n
p
−1 xnqα dμq x
p
1 − xqα
n
2.4
dμq x.
p
From 2.3 and 2.4, we have
1 −
xnq−α dμq x
p
n
n
α
α
α
−1l βl,q qαn −1n βn,q −1 βn,q 2.
l
l0
2.5
Therefore, we obtain the following lemma.
Lemma 2.1. For n ∈
,
one has
1 −
xnq−α dμq x
n
n
l0
p
l
α
α
α
−1l βl,q qαn −1n βn,q −1 βn,q 2,
2.6
α
α
βn,q−1 1 − x −1n qαn βn,q x.
By 2.2, 2.3, and 2.4, we get
α 1α
α
q q2 − q βn,q ,
αq
if n > 1.
2.7
nα α−1
1
1 α
α
q 1− ,
βn,q 2 2 βn,q q
αq
q
if n > 1.
2.8
q2 βn,q 2 n
α
Thus, we have
Therefore, by 2.8, we obtain the following proposition.
Journal of Inequalities and Applications
5
Proposition 2.2. For n ∈ with n > 1, one has
nα α−1
1
1 α
α
βn,q 2 2 βn,q q 1− .
q
αq
q
2.9
By using Proposition 2.2 and Lemma 2.1, we obtain the following corollary.
Corollary 2.3. For n ∈ with n > 1, one has
1 − xnq−α dμq x q2 βn,q−1 α
p
1 − xnq−α dμq x p
nα
1 − q q2
αq
Taking the bosonic q-integral on
we have
α Bk,n x, q dμq x
p
p
n
dμq x.
2.11
xkqα 1 − xn−k
q−α dμq x
p
n−k
n n−k
l0
l
l0
l
n−k
n n−k
k
1 − xqα
2.10
for one weighted q-Bernstein polynomials in 2.1,
k
k
xnq−α dμq−1 x n
p
p
nα
1 − q,
αq
l
xkl
qα dμq x
−1
2.12
p
−1l βkl,q .
α
By the symmetry of q-Bernstein polynomials, we get
α Bk,n x, q dμq x p
1
α
dμq x
Bn−k,n 1 − x,
q
p
k n k
kl
−1
k l0 l
1 − xn−l
q−α dμq x.
p
2.13
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Journal of Inequalities and Applications
For n > k 1, by 2.11 and 2.13, we have
α Bk,n x, q dμq x
p
k
k
n kl
nα
1 − q q2
αq
−1
k l0 l
⎧ nα
α
⎪
1 − q q2 βn,q−1 ,
⎪
⎪
⎪ αq
⎪
⎨
⎛ ⎞ k ⎛ ⎞
n
k
⎪
⎪
⎪
⎝ ⎠q2 ⎝ ⎠−1kl βα −1
⎪
⎪
n−l,q ,
⎩ k
l
l0
xn−l
q−α dμq−1 x
p
if k 0,
2.14
if k > 0.
By comparing the coefficients on the both sides of 2.12 and 2.14, we obtain the
following theorem.
Theorem 2.4. For n, k ∈
with n > k 1, one has
n−k
n−k
l0
l
α
−1l βkl,q
k
k
α
q
−1kl βn−l,q−1 ,
l
l0
2
if k / 0.
2.15
In particular, when k 0, one has
n
n
nα
α
2 α
1 − q q βn,q−1 −1l βl,q .
αq
l
l0
Let m, n, k ∈
2.16
with m n > 2k 1. Then we see that
α α Bk,n x, q Bk,m x, q dμq x
p
n
m
k
k
nm−2k
dμq x
x2k
qα 1 − xq−α
p
2k n
m 2k
k
k
l0
l
2k n
m 2k
k
k
l0
l
2k
n
m 2k
k
k
l0
l
l2k
dμq x.
1 − xnm−l
q−α
−1
2.17
p
l2k
−1
l2k
−1
nα
1 − q q2
αq
dμq−1 x
xnm−l
q−α
p
nα
α
1 − q q2 βnm−l,q−1
αq
Therefore, by 2.17, we obtain the following theorem.
.
Journal of Inequalities and Applications
Theorem 2.5. For m, n, k ∈
7
with m n > 2k 1, one has
⎧ nα
α
⎪
1 − q q2 βnm,q−1 ,
⎪
⎪
α
⎪
⎨⎛ q⎞⎛ ⎞
α ⎛ ⎞
α Bk,n x, q Bk,m x, q dμq x 2k
2k
n
m
⎪
⎪
p
⎝ ⎠⎝ ⎠q2 ⎝ ⎠−1l2k βα
⎪
,
⎪
nm−l,q−1
⎩
l
k
k
l0
if k 0,
if k /
0.
2.18
For m, n, k ∈
,
we have
α α Bk,n x, q Bk,m x, q dμq x
p
n
m
nm−2k
dμq x
x2k
qα 1 − xq−α
k
k
p
n
m nm−2k
n m − 2k
l
−1
x2kl
qα dμq x
k
k
l
p
l0
n m − 2k
n
m nm−2k
α
−1l βl2k,q .
l
k
k
l0
2.19
Therefore, by 2.18 and 2.19, we obtain the following theorem.
Theorem 2.6. For m, n, k ∈
with m n > 2k 1, one has
nm
nm
nα
α
2 α
1 − q q βnm−l,q−1 −1l βl,q .
αq
l
l0
2.20
Furthermore, for k / 0, one has
nm−2k
n m − 2k
l0
l
α
−1l βl2k,q
2k
2k
α
q
−1l2k βnm−l,q−1 .
l
l0
2.21
2
By the induction hypothesis, we obtain the following theorem.
Theorem 2.7. For s ∈ and k, n1 , . . . , ns ∈
with n1 n2 · · · ns > sk 1, one has
⎧ nα
α
⎪
1 − q q2 βn ···n ,q−1 ,
⎪
⎪
1
s
α
⎪
s
⎨ ⎛ q ⎛ ⎞⎞ ⎛ ⎞
α Bk,ni x, q dμq x s
sk
ni
sk
⎪
⎪
p
⎝ ⎝ ⎠⎠ ⎝ ⎠−1lsk βα
i1
⎪
,
⎪
n1 ···ns −l,q−1
⎩
k
l
i1
l0
if k 0,
if k /
0.
2.22
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Journal of Inequalities and Applications
For s ∈ , let k, n1 , . . . , ns ∈
with n1 n2 · · · ns > sk 1. Then we show that
s n ···n −sk
s
1 s
ni
n1 · · · ns − sk
α α
Bk,ni x, q dμq x −1l βlsk,q .
k
l
p
i1
i1
l0
2.23
Therefore, by Theorem 2.7 and 2.23, we obtain the following theorem.
Theorem 2.8. For s ∈ , let k, n1 , . . . , ns ∈
for k 0
n1 ···n
s
l0
with n1 n2 · · · ns > sk 1. Then one sees that
n1 · · · ns
nα
α
α
1 − q q2 βn ···n ,q−1 .
−1l βl,q 1
s
α
l
q
2.24
For k /
0, one has
sk
sk
l0
l
α
−1lsk βn1 ···ns −l,q−1
n1 ···n
s −sk
l0
n1 · · · ns − sk
α
−1l βlsk,q .
l
2.25
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