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.. Simsek, “
Trang 1Volume 2011, Article ID 513821, 8 pages
doi:10.1155/2011/513821
Research Article
A Study on the p-Adic q-Integral Representation on
q-Bernoulli Polynomials
T Kim,1 A Bayad,2 and Y.-H Kim1
1 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2 D´epartement de Math´ematiques, Universit´e d’Evry Val d’Essonne, Boulevard Franc¸ois 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
Copyrightq 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
of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic
assume|q| < 1 If q ∈p, we assume that|1 − q| p < 1 In this paper, we define the q-number
asx q 1 − q x /1 − q see 1 13
weighted q-Bernstein operator of order n for f ∈ C0, 1 is defined by
α
n,q
f | x
n
k0
f
k n
k
x k
q α 1 − x n−k
q −α n
k0
f
k n
B α k,n
x, q
Here B α k,n x, q is called the weighted q-Bernstein polynomials of degree n see 2,5,6
Trang 2Let UD p be the space of uniformly differentiable functions on p For f ∈ UD p,
I q
f
p
f xdμ q x lim
N → ∞
1
p N
q
pN−1
x0
see 10
The Carlitz’s q-Bernoulli numbers are defined by
β 0,q 1, qqβ 1k
− β k,q
with the usual convention about replacing β k by β k,qsee 3,9,10 In 3, Carlitz also defined
the expansion of Carlitz’s q-Bernoulli numbers as follows:
β h
0,q h
h q , q h qβ h 1
n
− β h n,q
1, if n 1,
with the usual convention about replacingβ hn
by β h n,q
for α ∈,
β α
0,q 1, q q α β α 1n − β α n,q
⎧
⎨
⎩
α
α q , if n 1,
1.5
with the usual convention about replacing β αn by β n,q α Let f n x fx n By the
definition1.2 of p-adic q-integral on p, we easily get
qI q
f1
q lim
N → ∞
1
p N
q
pN−1
x0
f x 1q x ,
lim
N → ∞
1
p N
q
pN−1
x0
f xq x lim
N → ∞
f
p N
q p N
− f0
p N
q
p
f xdμ q x q − 1
f0 q − 1
log q f
0,
1.6
Continuing this process, we obtain easily the relation
q n
p
f n xdμ q x −
p
f xdμ q x q − 1n−1
l0
q l f l q − 1
log q
n−1
l0
q l fl, 1.7
where n ∈ and fl dfl/dx see 6
Trang 3Then by1.2, applying to the function x → x n
q α, we can see that
β α
n,q
p
x n
q α dμ q x − α nα
q
∞
m0
q mαm m n−1
q α 1− q∞
m0
q m m n
The weighted q-Bernoulli polynomials are also defined by the generating function as
follows:
F q α t, x −t α α
q
∞
m0
q mαm e mx qα t1− q∞
m0
q m e mx qα t∞
n0
β α
n,q x t n
see6 Thus, we note that
β α
n,q x n
l0
n l
x n−l
q α q αlx β α
l,q
−α nα
q
∞
m0
q mαm m x n−1
q α 1− q∞
m0
q m m x n
q α
1.10
q-Bernoulli polynomials as follows:
β α
n,q x
p
x y n
q α dμ q
y
l0
n l
q αlx x n−l
q α
p
y l
q α dμ q
y
q n β α
m,q n − β α m,qq − 1n−1
l0
q l l m
q αα mα
q
n−1
l0
q αll l m−1
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 Bernoulli polynomials by using the expression of p-adic
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
k
x k
q α 1 − x n−k
q −α n
k0
f
k n
B k,n α
x, q
Trang 4The p-adic q-Bernstein polynomials with weight α of degree n are given by
B α k,n
x, q
n k
x k
q α 1 − x n−k
where x ∈ p , α ∈, and n, k ∈ see 6,7 Note that B α k,n x, q B n−k,n α 1 − x, 1/q That
is, the weighted q-Bernstein polynomials are symmetric.
From the definition of the weighted q-Bernoulli polynomials, we have
β α
n,q−11 − x −1 n q αn β α
By the definition of p-adic q-integral on p, we get
p
1 − x n
q −α dμ q x q αn−1n
p
−1 x n
q α dμ q x
p
1− x q α
n
dμ q x.
2.4
From2.3 and 2.4, we have
p
1 − x n
q −α dμ q x n
l0
n l
−1l β α
l,q q αn−1n β α
n,q −1 β α n,q 2. 2.5
Therefore, we obtain the following lemma
Lemma 2.1 For n ∈ , one has
p
1 − x n
q −α dμ q x n
l0
n l
−1l β α
l,q q αn−1n β α
n,q −1 β n,q α 2,
β α
n,q−11 − x −1 n q αn β α
n,q x.
2.6
By2.2, 2.3, and 2.4, we get
q2β α
n,q 2 n α
Thus, we have
β α
n,q2 1
q2β α
n,qα nα
q
q α−1 1 − 1
Therefore, by2.8, we obtain the following proposition
Trang 5Proposition 2.2 For n ∈with n > 1, one has
β α
n,q2 1
q2β α
n,q nα
α q q α−1 1 −
1
Corollary 2.3 For n ∈with n > 1, one has
p
1 − x n
q −α dμ q x q2β α
n,q−1α nα
q
p
1 − x n
q −α dμ q x α nα
q
1 − q q2
p
x n
q −α dμ q−1x
p
1− x q α
n
dμ q x.
2.11
we have
p
B α k,n
x, q
dμ q x
n k
p
x k
q α 1 − x n−k
q −α dμ q x
n k
n−k
l0
n − k l
−1l
p
x kl
q α dμ q x
n k
n−k
l0
n − k l
−1l β α
kl,q
2.12
By the symmetry of q-Bernstein polynomials, we get
p
B k,n α
x, q
dμ q x
p
B n−k,n α
1− x,1
q
dμ q x
n k
l0
k l
−1kl
1 − x n−l
q −α dμ q x.
2.13
Trang 6For n > k 1, by 2.11 and 2.13, we have
p
B k,n α
x, q
dμ q x
n k
l0
k l
−1kl
nα
α q 1 − q q2
p
x n−l
q −α dμ q−1x
⎧
⎪
⎪
⎪
⎪
nα
α q 1 − q q2β α
⎛
k
⎞
⎠q2k
l0
⎛
l
⎞
⎠−1kl β α
n−l,q−1, if k > 0.
2.14
following theorem
Theorem 2.4 For n, k ∈ with n > k 1, one has
n−k
l0
n − k l
−1l β α
kl,q q2k
l0
k l
−1kl β α
n−l,q−1, if k / 0. 2.15
In particular, when k 0, one has
nα
α q 1 − q q2β α
n,q−1 n
l0
n l
−1l β α
Let m, n, k ∈ with m n > 2k 1 Then we see that
p
B α k,n
x, q
B α k,m
x, q
dμ q x
n
k
m k
p
x 2k
q α 1 − x nm−2k
q −α dμ q x
n
k
m k
2k
l0
2k
l
−1l2k
p
1 − x nm−l
q −α dμ q x.
n
k
m k
2k
l0
2k
l
−1l2k
nα
α q 1 − q q2
p
x nm−l
q −α dμ q−1x
n
k
m k
l0
2k
l
−1l2k
nα
α q 1 − q q2β α
nm−l,q−1
.
2.17
Therefore, by2.17, we obtain the following theorem
Trang 7Theorem 2.5 For m, n, k ∈ with m n > 2k 1, one has
p
B α k,n
x, q
B k,m α
x, q
dμ q x
⎧
⎪
⎪
⎪
⎪
nα
α q 1 − q q2β α
nm,q−1, if k 0,
⎛
k
⎞
⎠
⎛
k
⎞
⎠q22k
l0
⎛
l
⎞
⎠−1l2k β α
nm−l,q−1, if k / 0.
2.18
p
B α k,n
x, q
B α k,m
x, q
dμ q x
n k
m k
p
x 2k
q α 1 − x nm−2k
q −α dμ q x
n k
m k
nm−2k
l0
n m − 2k l
−1l
p
x 2kl
q α dμ q x
n k
m k
nm−2k
l0
n m − 2k l
−1l β α
l2k,q
2.19
Therefore, by2.18 and 2.19, we obtain the following theorem
Theorem 2.6 For m, n, k ∈ with m n > 2k 1, one has
nα
α q 1 − q q2β α
nm−l,q−1 nm
l0
n m l
−1l β α
Furthermore, for k / 0, one has
nm−2k
l0
n m − 2k l
−1l β α
l2k,q q22k
l0
2k
l
−1l2k β α
nm−l,q−1. 2.21
By the induction hypothesis, we obtain the following theorem
Theorem 2.7 For s ∈ and k, n1, , n s∈ with n1 n2 · · · n s > sk 1, one has
p
i1
B k,n α i
x, q
dμ q x
⎧
⎪
⎪
⎪
⎪
nα
α q 1 − q q2β α
n1···n s ,q−1, if k 0,
⎛
i1
⎛
⎝n i
k
⎞
⎠
⎞
⎠sk
l0
⎛
l
⎞
⎠−1lsk β α
n1···n s −l,q−1, if k / 0.
2.22
Trang 8For s ∈, let k, n1, , n s∈ with n1 n2 · · · n s > sk 1 Then we show that
p
i1
B α k,n i
x, q
dμ q x
i1
n i
k
1···ns −sk
l0
n1 · · · n s − sk
l
−1l β α
lsk,q
2.23
Theorem 2.8 For s ∈, let k, n1, , n s∈ with n1 n2 · · · n s > sk 1 Then one sees that for k 0
n1···n s
l0
n1 · · · n s
l
−1l β α
l,q α nα
q
1 − q q2β α
n1···n s ,q−1. 2.24
For k / 0, one has
sk
l0
sk
l
−1lsk β α
n1···n s −l,q−1n1···ns −sk
l0
n1 · · · n s − sk
l
−1l β α
lsk,q 2.25
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... following proposition < /p> Trang 5Proposition 2.2 For n ∈with n > 1, one has< /p>
β... n ∈ and fl dfl/dx see 6 < /p> Trang 3
Then by1.2, applying to the. .. obtain the following theorem < /p> Trang 7
Theorem 2.5 For m, n, k ∈ with