Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2010, Article ID 179430, 9 pages
doi:10.1155/2010/179430
Research Article
On p-Adic Analogue of q-Bernstein Polynomials
and Related Integrals
T. Kim,1 J. Choi,1 Y. H. Kim,1 and L. C. Jang2
1
2
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Department of Mathematics and Computer Science, KonKuk University,
Chungju 380-701, Republic of Korea
Correspondence should be addressed to T. Kim, tkkim@kw.ac.kr
Received 17 September 2010; Accepted 22 December 2010
Academic Editor: Binggen Zhang
Copyright q 2010 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’s work in press introduced q-Bernstein polynomials which are different Phillips’
q-Bernstein polynomials introduced in the work by Phillips, 1996; 1997. The purpose of this
paper is to study some properties of several type Kim’s q-Bernstein polynomials to express the
p-adic q-integral of these polynomials on Zp associated with Carlitz’s q-Bernoulli numbers and
polynomials. Finally, we also derive some relations on the p-adic q-integral of the products of
several type Kim’s q-Bernstein polynomials and the powers of them on Zp .
1. Introduction
Let C0, 1 denote the set of continuous functions on 0, 1. For 0 < q < 1 and f ∈ C0, 1, Kim
introduced the q-extension of Bernstein linear operator of order n for f as follows:
Bn,q
n n
n k
k
k
n−k
Bk,n x, q ,
f
f
f |x xq 1 − x1/q n
n
k
k0
k0
1.1
where xq 1 − qx /1 − q see 1. Here Bn,q f | x is called Kim’s q-Bernstein operator
of order n for f. For k, n ∈ Z N ∪ {0}, Bk,n x, q nk xkq 1 − xn−k
1/q are called the Kim’s
q-Bernstein polynomials of degree n see 2–6.
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In 7, Carlitz defined a set of numbers ξk ξk q inductively by
⎧
⎨1
k
qξ 1 − ξk ⎩0
ξ0 1,
if k 1,
if k > 1,
1.2
with the usual convention of replacing ξk by ξk . These numbers are q-analogues of ordinary
Bernoulli numbers Bk , but they do not remain finite for q 1. So he modified the definition
as follows:
⎧
⎨1
k
q qβ 1 − βk,q ⎩0
β0,q 1,
if k 1,
if k > 1,
1.3
with the usual convention of replacing βk by βk,q see 7. These numbers βn,q are called the
nth Carlitz q-Bernoulli numbers. And Carlitz’s q-Bernoulli polynomials are defined by
βk,q x q β xq
x
k
k
k
βi,q qix xk−i
q .
i
i0
1.4
As q → 1, we have βk,q → Bk and βk,q x → Bk x, where Bk and Bk x are the ordinary
Bernoulli numbers and polynomials, respectively.
Let p be a fixed prime number. Throughout this paper, Z, Q, Zp , Qp , and Cp will denote
the ring of rational integers, the field of rational numbers, the ring of p-adic integers, the field
of p-adic rational numbers and the completion of algebraic closure of Qp , respectively. Let νp
be the normalized exponential valuation of Cp such that |p|p p−νp p 1/p.
Let q be regarded as either a complex number q ∈ C or a p-adic number q ∈ Cp . If
q ∈ C, we assume |q| < 1, and if q ∈ Cp , we normally assume |1 − q|p < 1.
We say that f is a uniformly differentiable function at a point a ∈ Zp and denote this
property by f ∈ UDZp if the difference quotient Ff x, y fx − fy/x − y has a limit
f a as x, y → a, a see 1, 3, 8–13.
For f ∈ UDZp , let us begin with the expression
1
qx fx fxμq x pN Zp ,
N
p q 0≤x<pN
0≤x<pN
1.5
representing a q-analogue of the Riemann sums for f see 11. The integral of f on Zp is
defined as the limit as n → ∞ of the sums if exists. The p-adic q-integral on a function
f ∈ UDZp is defined by
Iq f see 11.
Zp
1
fxdμq x lim N N →∞ p
q
N
−1
p
fxqx ,
x0
1.6
Discrete Dynamics in Nature and Society
3
As was shown in 3, Carlitz’s q-Bernoulli numbers can be represented by p-adic
q-integral on Zp as follows:
Zp
xm
q dμq x βm,q ,
for m ∈ Z .
1.7
Also, Carlitz’s q-Bernoulli polynomials βk,q x can be represented
βm,q x Zp
m
x y q dμq y ,
for m ∈ Z ,
1.8
see 3.
In this paper, we consider the p-adic analogue of Kim’s q-Bernstein polynomials on Zp
and give some properties of the several type Kim’s q-Bernstein polynomials to represent the
p-adic q-integral on Zp of these polynomials. Finally, we derive some relations on the p-adic
q-integral of the products of several type Kim’s q-Bernstein polynomials and the powers of
them on Zp .
2. q-Bernstein Polynomials Associated with p-Adic q-Integral on Zp
In this section, we assume that q ∈ Cp with |1 − q|p < 1.
From 1.5, 1.7 and 1.8, we note that
n
n
qn
l1
,
1 − x −1l qlx l1
n−1
q
−1
l
Zp
q−1
l0
n
n
l1
1
n
.
x x1 q dμq x1 −1l qlx
n−1
1 − ql1
l
Zp
1−q
l0
x1 n1/q dμ1/q x1 2.1
By 2.1, we get
−1n qn
Zp
x x1 nq dμq x1 Zp
1 − x x1 n1/q dμ1/q x1 .
2.2
Therefore, we obtain the following theorem.
Theorem 2.1. For n ∈ Z , one has
Zp
1 − x x1 n1/q dμ1/q x1 −1n qn
Zp
x x1 nq dμq x1 .
2.3
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By the definition of Carlitz’s q-Bernoulli numbers and polynomials, we get
n
q2 βn,q 2 − n 1q2 q q qβ 1 βn,q
if n > 1.
2.4
Thus, we have the following proposition.
Proposition 2.2. For n ∈ N with n > 1, one has
βn,q 2 1
1
βn,q n 1 − .
q
q2
2.5
It is easy to show that
n
1 − xn1/q 1 − xq −1n qn x − 1nq .
2.6
Hence, we have
Zp
1 − xn1/q dμq x −1n qn
Zp
x − 1nq dμq x.
2.7
By 1.8, we get
Zp
1 − xn1/q dμq x −1n qn βn,q −1.
2.8
By Theorem 2.1 and 2.8, we see that
Zp
1 − xn1/q dμq x −1n qn βn,q −1 βn,1/q 2.
2.9
From 2.9 and Proposition 2.2, we have
Zp
1 − xn1/q dμq x βn,1/q 2 q2 βn,1/q n 1 − q.
2.10
By 1.7 and 2.10, we obtain the following theorem.
Theorem 2.3. For n ∈ N with n > 1, one has
Zp
1 − xn1/q dμq x q2
Zp
xn1/q dμ1/q x n 1 − q.
2.11
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Taking the p-adic q-integral on Zp for one Kim’s q-Bernstein polynomials, we get
Zp
Bk,n x, q dμq x n
k
Zp
xkq 1 − xn−k
1/q dμq x
n−k n − k
n k
l0
l
l0
l
−1l
Zp
xkl
q dμq x
2.12
n−k n − k
n k
−1l βkl,q ,
and, by the q-symmetric property of Bk,n x, q, we see that
Zp
Bk,n x, q dμq x 1
dμq x
Bn−k,n 1 − x,
q
Zp
k
k
n k
l0
−1
l
2.13
kl
Zp
1 −
xn−l
1/q dμq x.
For n > k 1, by Theorem 2.3 and 2.13, one has
Zp
Bk,n
k
k
n kl
n−l
2
n−l1−qq
x, q dμq x −1
x1/q dμ1/q x
l
k l0
Zp
k
n k kl
n − l 1 − q q2 βn−l,1/q .
−1
k l0
l
2.14
Let m, n, k ∈ Z with m n > 2k 1. Then the p-adic q-integral for the multiplication
of two Kim’s q-Bernstein polynomials on Zp can be given by the following relation:
Zp
Bk,n x, q Bk,m x, q dμq x n
m
k
k
Zp
nm−2k
dμq x
x2k
q 1 − x1/q
2k
2k
n
m 1 − xnm−l
−1l2k q
1/q dμq x.
k
k l0 l
Zp
2.15
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By Theorem 2.3 and 2.15, we get
Zp
Bk,n x, q Bk,m x, q dμq x
2k
2k
n
m k
k
l
l0
2k
n
m 2k
k
k
l
l0
−1
l2k
nm−l1−qq
xnm−l
1/q dμ1/q x
2
Zp
2.16
−1l2k n m − l 1 − q q2 βnm−l,1/q .
By the simple calculation, we easily get
Zp
Bk,n x, q Bk,m x, q dμq x n
m
nm−2k
dμq x
x2k
q 1 − x1/q
k
k
Zp
n
m nm−2k
n m − 2k
k
k
l0
l
l0
l
−1l
Zp
dμq x
xl2k
q
n
m nm−2k
n m − 2k
k
k
−1l βl2k,q .
2.17
Continuing this process, we obtain
s
Zp
Bk,ni x, q
dμq x i1
s
ni
i1
k
Zp
n1 ···ns −sk
dμq x
xsk
q 1 − x1/q
s sk
ni
sk
i1
k
l0
l
−1
skl
Zp
1 ···ns −l
dμq x.
1 − xn1/q
2.18
Let s ∈ N and n1 , . . . , ns , k ∈ Z with n1 n2 · · · ns > sk 1. By Theorem 2.3 and
2.18, we get
s
Bk,ni x, q dμq x
Zp
i1
s sk
ni
sk
i1
k
l0
l
s
sk
ni
sk
i1
k
l0
l
−1
skl
s
2
ni − l 1 − q q
i1
−1
skl
Zp
1 ···ns −l
dμ1/q x
xn1/q
s
2
ni − l 1 − q q βn1 ···ns −l,1/q .
i1
2.19
Discrete Dynamics in Nature and Society
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From the definition of binomial coefficient, we note that
s
Bk,ni x, q dμq x
Zp
i1
s ni
k
i1
Zp
n1 ···ns −sk
dμq x
xsk
q 1 − x1/q
n ···n −sk
s
1 s
ni
n1 · · · ns − sk
l
dμq x
−1
xskl
q
l
k
Zp
i1
l0
n ···n −sk
s
1 s
ni
n1 · · · ns − sk
−1l βskl,q ,
l
k
i1
l0
2.20
where s ∈ N and n1 , . . . , ns , k ∈ Z .
By 2.19 and 2.20, we obtain the following theorem.
Theorem 2.4. I For s ∈ N and n1 , . . . , ns , k ∈ N with n1 n2 · · · ns > sk 1, one has
s
Bk,ni x, q dμq x
Zp
i1
s
sk
ni
sk
k
i1
l0
s
skl
2
ni − l 1 − q q βn1 ···ns −l,1/q .
−1
l
2.21
i1
II For s ∈ N and n1 , . . . , ns , k ∈ Z , one has
s
Zp
Bk,ni x, q
dμq x i1
n ···n −sk
s
1 s
ni
n1 · · · ns − sk
i1
k
l0
−1l βskl,q .
l
2.22
By Theorem 2.4, we obtain the following corollary.
Corollary 2.5. For s ∈ N and n1 , . . . , ns , k ∈ N with n1 n2 · · · ns > sk 1, one has
sk
sk
l0
l
−1
skl
s
ni − l 1 − q q2 βn1 ···ns −l,1/q
i1
n1 ···n
s −sk n1 · · · ns − sk
−1l βskl,q .
l
l0
2.23
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Let s ∈ N and m1 , . . . , ms , n1 , . . . , ns , k ∈ Z with m1 n1 · · · ms ns > m1 · · · ms k 1.
Then one has
⎛ ⎞
s
mi k s mi
i1
s
⎜k mi ⎟
ni
mi Bk,n
x, q dμq x ⎝ i1 ⎠−1k i1 mi −l
i
k
i1
i1
l0
l
s
n m −l
×
1 − xq i1 i i dμq x
s
Zp
s
Zp
⎛ ⎞
s
mi k s mi
s
i1
s
m
k
ni
i⎟
⎜
⎝ i1 ⎠−1k i1 mi −l
k
i1
l0
l
s
s
2
i1 ni mi −l
×
mi ni − l 1 − q q
dμ1/q x
x1/q
i1
Zp
⎛ ⎞
s
s mi k s mi
i1
s
ni
m
k
i
⎜
⎟
⎝ i1 ⎠−1k i1 mi −l
k
i1
l0
l
s
2
×
mi ni − l 1 − q q βn1 m1 ···ns ms −l,1/q .
i1
2.24
From the definition of binomial coefficient, one has
s
Zp
i1
mi Bk,n
x, q
i
dμq x
⎛ s
⎞
s
mi s ni mi −k s mi s
i1
i1 ⎜ ni mi − k mi ⎟
ni
⎝ i1
⎠−1l
i1
k
i1
l0
l
1 ···ms kl
×
dμq x
xm
q
Zp
⎛ s
⎞
s
mi s ni mi −k s mi s
i1
i1 ⎜ ni mi − k mi ⎟
ni
⎝ i1
⎠
i1
k
i1
l0
l
× −1l βm1 ···ms kl,q .
By 2.24 and 2.25, we obtain the following theorem.
2.25
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Theorem 2.6. For s ∈ N and m1 , . . . , ms , n1 , . . . , ns , k ∈ Z with m1 n1 · · · ms ns > m1 · · · ms k 1, one has
k
⎛ ⎞
s
s
i1 mi
m
k
i⎟
⎜
k si1 mi −l
2
mi ni − l 1 − q q βn1 m1 ···ns ms −l,1/q
⎝ i1 ⎠−1
s
l0
i1
l
⎛
⎞
s
s
−k
m
i
i
i1 ni m
i1
⎜ ni mi − k mi ⎟
⎝ i1
⎠−1l βm1 ···ms kl,q .
i1
s
s
l0
2.26
l
Acknowledgment
This paper was supported by the research grant of Kwangwoon University in 2010.
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