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. Kim
1
1
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2
D
´
epartement de Ma th
´
ematiques, Universit
´
e d’Evry Val d’Essonne, Boulevard Franc¸ois Mitterrand,
91025 Evry Cedex, France
Correspondence should be addressed to A. Bayad, [email protected]
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
or a p-adic number q ∈
p
.Ifq ∈ ,thenwealways
assume |q| < 1. If q ∈
p
, we assume that |1 − q|
p
< 1. In this paper, we define the q-number
as x
q
1 − q
x
/1 − qsee 1–13.
Let C0, 1 be the set of continuous functions on 0, 1.Forα ∈
and n, k ∈
,the
weighted q-Bernstein operator of order n for f ∈ C0, 1 is defined by
α
n,q
f | x
n
k0
f
k
x, q
. 1.1
Here B
α
k,n
x, q is called the weighted q-Bernstein polynomials of degree n see 2, 5, 6.
2 Journal of Inequalities and Applications
Let UD
p
be the space of uniformly differentiable functions on
p
.Forf ∈ UD
p
,
the p-adic q-integral on
p
, which is called the bosonic q-integral on
p
,isdefinedby
I
q
f
p
f
β
0,q
1,q
qβ 1
k
− β
k,q
1, if k 1,
0, if k>1,
1.3
with the usual convention about replacing β
k
by β
k,q
see 3, 9, 10.In3, Carlitz also defined
the expan sion of Carlitz’s q-Bernoulli numbers as follows:
β
h
0,q
h
h
q
,q
1,q
q
α
β
α
1
n
−
β
α
n,q
⎧
⎨
⎩
α
α
q
, if n 1,
0, if n>1,
1.5
with the usual convention about replacing
β
−1
x0
f
x 1
q
x
,
lim
N →∞
1
p
N
q
p
N
−1
x0
f
x
q
x
lim
q − 1
f
0
q − 1
log q
f
0
,
1.6
Continuing this process, we obtain easily the relation
q
n
p
f
n
x
dμ
q
log q
n−1
l0
q
l
f
l
, 1.7
where n ∈
and f
ldf l/dx see 6.
Journal of Inequalities and Applications 3
Then by 1.2, applying to the function x → x
n
q
α
,wecanseethat
β
α
n,q
p
∞
m0
q
m
m
n
q
α
. 1.8
The weighted q-Bernoulli polynomials are also defined by the generating function as
follows:
F
α
q
t, x
−t
α
α
q
∞
m0
x
t
n
n!
,
1.9
see6. Thus, we note that
β
α
n,q
x
n
l0
n
l
x
n−l
q
α
q
m
m x
n
q
α
.
1.10
From 1.2 and the previous equalities, we obtain the Witt’s formula for the weighted
q-Bernoulli polynomials as follows:
β
α
n,q
x
p
x y
n
q
α
dμ
q
. 1.11
By using 1.2 and the weighted q-Bernoulli polynomials, we easily get
q
n
β
α
m,q
n
−
β
α
m,q
q − 1
n−1
l0
q
l
l
m
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 | x
fx
n
k0
f
k
n
n
k
α
k,n
x, q
n
k
x
k
q
α
1 − x
n−k
q
−α
, 2.2
where x ∈
p
, α ∈ ,andn, k ∈
see 6, 7.NotethatB
α
k,n
p
,weget
p
1 − x
n
q
−α
dμ
q
x
q
αn
−1
n
p
−1 x
n
q
α
dμ
q
−α
dμ
q
x
n
l0
n
l
−1
l
β
α
l,q
q
αn
−1
n
x
n
l0
n
l
−1
l
β
α
l,q
q
αn
−1
n
β
α
n,q
−1
x
.
2.6
By 2.2, 2.3,and2.4,weget
q
2
β
α
n,q
2
n
α
α
q
q
1α
q
2
− q
β
α
n,q
, if n>1. 2.7
Proposition 2.2. For n ∈
with n>1,onehas
β
α
n,q
2
1
q
2
β
α
n,q
nα
α
q
q
α−1
1 −
1
q
. 2.9
By using Proposition 2.2 and Lemma 2.1, we obtain the following corollary.
p
1 − x
n
q
−α
dμ
q
x
nα
α
q
1 − q q
2
p
x
n
q
−α
dμ
q
α
k,n
x, q
dμ
q
x
n
k
p
x
k
q
α
1 − x
n−k
q
−α
dμ
x
n
k
n−k
l0
n − k
l
−1
l
β
α
kl,q
.
2.12
By the symmetry of q-Bernstein polynomials, we get
p
B
α
k
l0
k
l
−1
kl
p
1 − x
n−l
q
−α
dμ
q
x
.
2.13
6 Journal of Inequalities and Applications
For n>k 1, by 2.11 and 2.13,wehave
p
α
q
1 − q q
2
p
x
n−l
q
−α
dμ
q
−1
x
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
k
l
⎞
⎠
−1
kl
β
α
n−l,q
−1
,
if k>0.
2.14
By comparing the coefficients on the both sides of 2.12 and 2.14,weobtainthe
following theorem.
Theorem 2.4. For n, k ∈
with n>k 1,onehas
n−k
l0
n − k
l
−1
α
q
1 − q q
2
β
α
n,q
−1
n
l0
n
l
−1
l
β
α
l,q
. 2.16
Let m, n, k ∈
with m n>2k 1. Then we see that
x
2k
q
α
1 − x
nm−2k
q
−α
dμ
q
x
n
k
m
k
2k
l0
2k
l
2k
l
−1
l2k
nα
α
q
1 − q q
2
p
x
nm−l
q
−α
dμ
q
−1
x
α
nm−l,q
−1
.
2.17
Therefore, by 2.17, we obtain the following theorem.
Journal of Inequalities and Applications 7
Theorem 2.5. For m, n, k ∈
with m n>2k 1,onehas
p
B
α
k,n
x, q
B
α
k,m
x, q
dμ
q
x
⎠
⎛
⎝
m
k
⎞
⎠
q
2
2k
l0
⎛
⎝
2k
l
⎞
⎠
−1
l2k
β
α
nm−l,q
−1
, if k
/
0.
p
x
2k
q
α
1 − x
nm−2k
q
−α
dμ
q
x
n
k
m
k
nm−2k
nm−2k
l0
n m − 2k
l
−1
l
β
α
l2k,q
.
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,onehas
nα
α
q
1 − q q
2
β
−1
l
β
α
l2k,q
q
2
2k
l0
2k
l
−1
l2k
β
α
nm−l,q
−1
. 2.21
By the induction hypothesis, we obtain the fo llowing theorem.
Theorem 2.7. For s ∈
and k, n
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
nα
α
q
1 − q q
2
β
α
n
1
···n
s
,q
−1
, if k 0,
n
1
···n
s
−l,q
−1
, if k
/
0.
2.22
8 Journal of Inequalities and Applications
For s ∈
,letk, n
1
, ,n
s
∈
with n
1
n
2
··· n
s
>sk 1. Then we show that
p
s
l0
n
1
··· n
s
− sk
l
−1
l
β
α
lsk,q
.
2.23
Therefore, by Theorem 2.7 and 2.23, we obtain the following theorem.
Theorem 2.8. For s ∈
,letk, n
1
, ,n
s
∈
with n
1
n
q
1 − q q
2
β
α
n
1
···n
s
,q
−1
. 2.24
For k
/
0,onehas
sk
l0
sk
l
−1
lsk
β
lsk,q
. 2.25
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