Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2010, Article ID 706483, 12 pages
doi:10.1155/2010/706483
Review Article
A Note on the Modified q-Bernstein Polynomials
Taekyun Kim,1 Lee-Chae Jang,2 and Heungsu Yi3
1
Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea
Department of Mathematics and Computer Science, Konkuk University,
Chungju 138-701, Republic of Korea
3
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2
Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr
Received 20 May 2010; Revised 11 July 2010; Accepted 14 July 2010
Academic Editor: Leonid Shaikhet
Copyright q 2010 Taekyun 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 propose the modified q-Bernstein polynomials of degree n which are different q-Bernstein
polynomials of Phillips 1997. From these modified q-Bernstein polynomials of degree n, we
derive some recurrence formulae for the modified q-Bernstein polynomials.
1. Introduction
Let C0, 1 denote the set of continuous function on 0, 1. For f ∈ C0, 1, Bernstein
introduced the following well-known linear positive operators in 1:
n n k
k
n k
f
f
Bn f : x :
Bk,n x,
x 1 − xn−k k
n
n
k0
k0
1.1
where nk nn − 1 · · · n − k 1/k!. Here Bn f : x is called the Bernstein operator of order n
for f. For k, n ∈ Z , the Bernstein polynomial of degree n is defined by
Bk,n x n k
x 1 − xn−k ,
k
1.2
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where x ∈ 0, 1. For example,
B0,1 x 1 − x,
B1,1 x x,
B0,2 x 1 − x2 , B1,2 x 2x1 − x, B2,2 x x2 , . . . .
1.3
Also, Bk,n x 0, for k > n, because nk 0.
Some people have studied the Bernstein polynomials in the area of approximation
theory see 2 through 3. Note that for k ∈ Z and x ∈ 0, 1,
tk e1−xt xk xk
k!
k!
t
∞
1 − xn tn
n!
n0
∞
xk 1 − xn n 1 · · · n k nk
t
k! n0
n k!
∞ n
k
nk
k
∞
x 1 − x
Bk,n x
nk
k
n−k
tn
n!
1.4
tn
.
n!
Because Bk,0 x Bk,1 x · · · Bk,k−1 x 0, we obtain the generating function for Bk,n x
as follows:
F k t, x :
∞
tn
tk e1−xt xk Bk,n x
k!
n!
n0
1.5
see 4, 5, where k ∈ Z and x ∈ 0, 1. Notice that
⎧⎛ ⎞
⎪
n
⎪
⎪
⎨⎝ ⎠xk 1 − xn−k
Bk,n x k
⎪
⎪
⎪
⎩
0
if n ≥ k,
1.6
if n < k,
for n, k ∈ Z see 2.
Let 0 < q < 1. Define the q-number of x by
xq :
1 − qx
.
1−q
1.7
See 2 through 3 for details and related facts. Note that limq → 1 xq x. In 6, Phillips
proposed a generalization of the classical Bernstein polynomials based on q-integers. In the
last decade some new generalizations of well-known positive linear operators, based on
q-integers were introduced and studied by several authors see 1–13. Recently, Simsek
Discrete Dynamics in Nature and Society
3
and Acikgoz have also studied the q-extension of Bernstein-type polynomials 5. Their qBernstein-type polynomials are given by
Yn k; x : q k
n−k ∞ kl−1
n −1 k! n−k
l
k 1 − qn−k m,l0 j0
k
×
m −1j qlj1−x Sm, k x ln q
,
m!
1.8
where Sm, k are the second-kind stirling number. In 5, we can find some interesting
formulae related to q-extension of Bernstein polynomials which are different q-Bernstein
polynomials of Phillips. In the conference of Jangjeon Mathematical Society which was held
in IRAN on Feb.2010, Acikgoz and Arci has introduced several-type Bernstein polynomials
see 2. The Acikgoz paper 2 announced in the conference is actually what motivated us to
write this paper. In this paper, we considered the q-extension of Bernstein polynomials which
were introduced by Acikgoz at the conference of Jangjeon Mathematical Society on Feb. 2010.
First, we consider the q-extension of the generating function of Bernstein polynomials in 1.5.
Indeed, this generating function is also treated by Simsek and Acikgoz in a previous paper
see 5. From this q-extension of the generating function for the Bernstein polynomials, we
propose the modified q-Bernstein polynomials of degree n which are different q-Bernstein
polynomials of Phillips. By using the properties of the modified q-Bernstein polynomials, we
obtain some recurrence formulae for the modified q-Bernstein polynomials of degree n.
2. The Modified q-Bernstein Polynomials
For 0 < q < 1, consider the q-extension of 1.5 as follows:
k
Fq t, x :
tk e1−xq t xkq
k!
∞ 1
xkq k!
nk
n!
n0
∞ n
k
− xnq
2.1
tnk
xkq 1 − xn−k
q
tn
,
n!
k
where k, n ∈ Z and x ∈ 0, 1. Note that limq → 1 Fq t, x F k t, x. We define the modified
q-Bernstein polynomials as follows:
k
Fq t, x
where k, n ∈ Z and x ∈ 0, 1.
tk e1−xq t xkq
k!
∞
tn
,
Bk,n x, q
n!
n0
2.2
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Remark. This generating function is also introduced by Simsek and Acikgoz in a previous
paper see 5.
By comparing the coefficients of 2.1 and 2.2, we obtain the following theorem.
Theorem 2.1. For k, n ∈ Z and x ∈ 0, 1,
⎧⎛ ⎞
⎪
n
⎪
k
n−k
⎪
⎨⎝ ⎠xq 1 − xq ,
Bk,n x, q k
⎪
⎪
⎪
⎩
0,
if n ≥ k
2.3
if n < k.
For 0 ≤ k ≤ n, we have
1 − xq Bk,n−1 x, q xq Bk−1,n−1 x, q
n−1
n−1
n−k
x
xkq 1 − xn−1−k
xk−1
q
q 1 − xq
q
k
k−1
n−1
n−1
k
n−k
xq 1 − xq xkq 1 − xn−k
q
k
k−1
n
xkq 1 − xn−k
q ,
k
1 − xq
2.4
and the derivatives of the modified q-Bernstein polynomials of degree n are also polynomials
of degree n − 1, that is,
n
n
n−k ln q x
k
n−k−1 − ln q
q
q1−x
−
x
−
k1
−
x
kxk−1
1
n
x
q
q
q
q
k
k
q−1
q−1
ln q
n
n
k−1
n−k x
k
n−k−1 1−x
q
kxq 1 − xq q −
xq n − k1 − xq
k
k
q−1
d
Bk,n x, q dx
ln q
n qx Bk−1,n−1 x, q − q1−x Bk,n−1 x, q
.
q−1
2.5
Therefore, we obtain the following recurrence formulae.
Theorem 2.2 recurrence formulae for Bk,n x, q. For k, n ∈ Z and for x ∈ 0, 1,
1 − xq Bk,n−1 x, q xq Bk−1,n−1 x, q Bk,n x, q ,
ln q
d
Bk,n x, q n qx Bk−1,n−1 x, q − q1−x Bk,n−1 x, q
.
dx
q−1
2.6
Discrete Dynamics in Nature and Society
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Let f be a continuous function on 0, 1. Then the modified q-Bernstein operator of order
n for f is defined by
n k
Bk,n x, q ,
f
Bn,q f : x :
n
k0
2.7
where 0 ≤ x ≤ 1, n ∈ Z . We get from Theorem 2.1 and 2.7 that for fx x,
Bn,q
n k
n
f
f :x xkq 1 − xn−k
q
k
n
k0
n−1
xq 1 − 1 − xq xq q − 1
2.8
n−1
f xq 1 1 − qxq 1 − xq
.
We also see from Theorem 2.1 that
Bn,q 1 : x n
Bk,n x, q
k0
n n
xkq 1 − xn−k
q
k
k0
n k0
n−k
n
xkq 1 − q1−x xq
k
2.9
n
1 1 − q xq 1 − xq .
The modified q-Bernstein polynomials are symmetric polynomials in the following
sense:
Bn−k,n 1 − x, q n
k
1 − xn−k
q xq Bk,n x, q .
n−k
2.10
Therefore, we get the following theorem.
Theorem 2.3. For k, n ∈ Z and x ∈ 0, 1,
Bn−k,n 1 − x, q Bk,n x, q ,
n
Bn,q 1 : x 1 1 − q xq 1 − xq .
2.11
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For ζ ∈ C, x ∈ 0, 1 and for n ∈ Z , consider
n!
2πi
xq ζ
C
k!
k
e1−xq ζ
2.12
dζ
,
ζn1
where C is a circle around the origin and integration is in the positive direction. We see
from the definition of the modified q-Bernstein polynomials and the basic theory of complex
analysis including Laurent series that
xq ζ
C
k
e
k!
1−xq ζ
m
∞ B
Bk,n x, q
dζ
dζ
k,m x, q ζ
.
2πi
m!
n!
ζn1 m0 C
ζn1
2.13
We get from 2.12 and 2.13 that
n!
2πi
xq ζ
e1−xq ζ
k!
C
k
xq ζ
k
e
k!
C
1−xq ζ
dζ
Bk,n x, q ,
n1
ζ
2.14
∞
1 − xm
xkq dζ
q
m−n−1k
ζ
dζ
k! m0
m!
ζn1
C
⎛
2πi⎝
xkq 1
−
xn−k
q
k!n − k!
⎞
2.15
⎠.
We also get from 2.12 and 2.15 that
n!
2πi
C
xq ζ
k!
k
e1−xq ζ
dζ
ζn1
n
xkq 1 − xn−k
q .
k
2.16
Therefore, we see from 2.14 and 2.16 that
Bk,n x, q n
xkq 1 − xn−k
q .
k
2.17
Discrete Dynamics in Nature and Society
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Note that
n−k
k1
Bk,n x, q Bk1,n x, q
n
n
n − 1!
n − 1!
n−k−1
xkq 1 − xn−k
xk1
q
q 1 − xq
k!n − k − 1!
k!n − k − 1!
1 − xq xq Bk,n−1 x, q
2.18
1 xq 1 − q1−x Bk,n−1 x, q
1 1 − q xq 1 − xq Bk,n−1 x, q .
Therefore, we can write the modified q-Bernstein polynomials as a linear combination of
polynomials of higher order as follows.
Theorem 2.4. For k, n ∈ Z and x ∈ 0, 1,
n1−k
k1
Bk,n1 x, q Bk1,n1 x, q 1 1 − q xq 1 − xq Bk,n x, q .
n1
n1
2.19
We easily see from 2.17 that for n, k ∈ N,
n−k1
k
x q
1 − xq
Bk−1,n x, q n−k1
k
x q
n
n−k1
xk−1
q 1 − xq
k−1
1 − xq
n!
xkq 1 − xn−k
q
k!n − k!
Bk,n x, q .
2.20
Thus, the following corollary holds.
Corollary 2.5. For n, k ∈ N and x ∈ 0, 1,
n−k1
k
x q
1 − xq
Bk−1,n x, q Bk,n x, q .
2.21
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Note from the definition of the modified q-Bernstein polynomials and the binomial
theorem that for k, n ∈ Z ,
Bk,n
n
x, q xkq 1 − xn−k
q
k
n−k
n
xkq 1 − q1−x xq
k
n−k n
n−k
xkq
−1l ql1−x xlq
k
l
l0
2.22
n−k kl
n
−1l ql1−x xlk
q
k
kl
l0
n n
n
jk
k
j
j
−1j−k q1−xj−k xq .
Therefore, we showed that the following theorem holds.
Theorem 2.6. For k, n ∈ Z and x ∈ 0, 1,
n j
n
j
Bk,n x, q −1j−k q1−xj−k xq .
k
j
jk
2.23
It is possible to write xkq as a linear combination of the modified q-Bernstein
polynomials by using the degree evaluation formulae and mathematical induction. We easily
see from the property of the modified q-Bernstein polynomials that
n k
k1
n n−1
Bk,n x, q xkq 1 − xn−k
q
k
−
1
n
k1
n−1 n−1
k0
k
n−1−k
xk1
q 1 − xq
2.24
n−1
xq xq 1 − xq
,
and that
n
k
2
n Bk,n x, q
k2 2
n n−2
xkq 1 − xn−k
q
k
−
2
k2
n−2 n−2
n−2−k
xk2
q 1 − xq
k
k0
n−2
x2q xq 1 − xq
.
2.25
Discrete Dynamics in Nature and Society
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Continuing this process, we obtain
n
kj
n−j
j
,
n Bk,n x, q xq xq 1 − xq
k
j
2.26
j
for j ∈ N. Therefore, we obtain the following theorem.
Theorem 2.7. For n, j ∈ Z and x ∈ 0, 1,
1
1 − xq xq
n−j
n
k
j
j
n Bk,n x, q xq .
j
kj
2.27
For k ∈ N, the Bernoulli polynomial of order k is defined by
t
et − 1
k
ext t
et − 1
× ··· ×
∞
tn
t
k
ext Bn x ,
t
n!
e −1
n0
2.28
k-times
k
k
and Bn Bn 0 are called the nth Bernoulli numbers of order k. It is well known that the
second kind stirling number is defined by
k
∞
et − 1
tn
:
Sn, k ,
k!
n!
n0
2.29
for k ∈ N. We note from 2.2 that
xq t
k
e1−xq t
k!
k
xkq et − 1 k
t
e1−xq t
k!
et − 1
∞
∞
tn
tm
k
k
xq
Sm, k
Bn 1 − xq
m!
n!
m0
n0
⎞
⎛
k
∞
l Bn
1 − xq Sl − n, kl! tl
⎟
⎜
xkq ⎝
⎠ .
n!l
−
n!
l!
n0
l0
2.30
We have from 2.2 and 2.30 that
Bk,l x, q xkq
l l
n0
n
k
Bn
and Bk,0 x, q Bk,1 x, q · · · Bk,k−1 x, q 0.
1 − xq Sl − n, k,
2.31
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Remark. The Equations 2.30 and 2.31 are already known by Simsek and Acikgoz in a
previous paper 5, page 7.
Let Δ be the shift difference operator defined by Δfx fx 1 − fx. We see from
the iterative method that
Δ f0 n
n n
k
k0
−1n−k fk,
2.32
for n ∈ N. We get from 2.29 and 2.32 that
∞
Sn, k
n0
k tn
1
k
−1k−l elt
n! k! l0 l
∞
k 1
k
n0
l
k! l0
−1
k−l n
l
tn
n!
2.33
∞
Δk 0n tn
.
k! n!
n0
By comparing the coefficients on both sides above, we have
Sn, k Δk 0n
,
k!
2.34
for n, k ∈ Z . Thus, we get from 2.31 and 2.34 that
Bk,l x, q xkq
l l
n0
n
k
Bn
1 − xq
Δk 0l−n
k!
.
2.35
Let Ehx hx 1 be the shift operator. Then the q-difference operator is defined by
j
Δnq Πn−1
j0 E − q I ,
2.36
where I is an identity operatorsee 7 through 11. For f ∈ C0, 1 and n ∈ N, we have
Δnq f0 n n
k0
k
n
−1k q 2 fn − k,
2.37
q
where nk q is the Gaussian binomial coefficient defined by
xq x − 1q · · · x − k 1q
x
.
k q
kq !
2.38
Discrete Dynamics in Nature and Society
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Let Fq t be the generating function of the q-extension of the second kind stirling number
as follows:
k−j
− k
∞
tn
q 2 k
.
q 2 ejq t S n, k : q
−1k−j
j q
n!
kq ! j0
n0
2.39
k
q− 2 n q− 2 k n
j k
S n, k : q Δ 0 ,
k−j q −1j q 2
j q
kq ! j0
kq ! q
2.40
k
Fq t :
We have from 2.39 that
k
k
where kq ! kq k − 1q · · · 2q 1q . It is not difficult to see that
xnq n
q
k
2
k0
x
k !S n, k : q .
k q q
2.41
See also 7 through 11 for details and related facts for above. Then, we get from 2.41 and
Theorem 2.7 that
j
k x
q 2
kq !S j, k : q k q
k0
1
1 − xq xq
n−j
n
kj
k
j
n Bk,n x, q .
j
2.42
Therefore, this completes the proof of the following theorem.
Theorem 2.8. For n, j ∈ Z and x ∈ 0, 1,
1
1 − xq xq
n−j
n
kj
k
j
j
k x
q 2
kq !S j, k : q .
n Bk,n x, q k q
j
k0
2.43
Acknowledgments
The authors express their sincere gratitude to the referees for their valuable suggestions and
comments. This paper is supported in part by the Research Grant of Kwangwoon University
in 2010.
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