Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 634675, 11 pages
doi:10.1155/2011/634675
Review Article
Some Identities on the q-Integral
Representation of the Product of Several
q-Bernstein-Type Polynomials
Taekyun Kim
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr
Received 28 August 2011; Accepted 5 November 2011
Academic Editor: Pavel Drábek
Copyright q 2011 Taekyun Kim. 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.
The purpose of this paper is to give some properties of several q-Bernstein-type polynomials to
express the q-integral on 0, 1 in terms of q-beta and q-gamma functions. Finally, we derive some
identities on the q-integral of the product of several q-Bernstein-type polynomials.
1. Introduction
Let q ∈ R with 0 ≤ q < 1. We assume that q-number is defined by xq 1 − qx /1 − q and
0q 0. Note that limq → 1 xq x. The q-derivative of a map f : R → R at x ∈ R \ {0} is
given by
dq fx f qx − fx
Dq f dq x
q−1 x
1.1
see 1–6. For n ∈ N, by 1.1, we get Dqn xn nq n−1q · · · 2q 1q n1 !. The q-binomial
formula is given by
a bnq
n−1 a bq
i0
i
n
n
l0
l
q
l
2 an−l bl
q
see 2, 5, 7–11, where nk q nq !/kq !n − kq ! nq n − 1q · · · n − k 1q /kq !.
1.2
2
Abstract and Applied Analysis
For a, b ∈ R, the Jackson q-integral of f : R → R is defined by
b
∞
fxdq x 1 − q
qn bf bqn − af aqn
a
1.3
n0
see 1, 2, 5, 6, 9, 12, 13. From 1.2, we note that
n1
k
n
q
k−1
q
k
n
k
q
q
n
n−k
k−1
q
n
q
k
.
1.4
q
By 1.2 and 1.4, we get
1 −
bnq
b:q
n
n−1 1−qb i
n
n
i0
i0
1
1
1
n−1 i
1 − bnq
b:q n
i0 1 − q b
i
i
q 2 −bi ,
q
1.5
∞
ni−1
i0
i
bi .
q
Let C0, 1 denote the set of continuous function on 0, 1. For f ∈ C0, 1, Bernstein
introduced the following well-known linear operators see 1, 4, 9, 11, 14:
n n
n k
k
xk 1 − xn−k Bn f | x f
f
Bk,n x.
n
n
k
k0
k0
1.6
Here Bn f | x is called Bernstein operator of order n for f. For k, n ∈ Z N ∪ {0}, the
Bernstein polynomials of degree n are defined by
Bk,n x n
k
xk 1 − xn−k
1.7
see 1, 3, 4, 11–14. By the definition of Bernstein polynomials see 1.6 and 1.7, we can
see that Bernstein basis is the probability mass function of binomial distribution. A Bernoulli
trial involves performing an experiment once and noting whether a particular event A occurs.
The outcome of Bernoulli trial is said to be “success” if A occurs and a “failure” otherwise.
Let k be the number of successes in n independent Bernoulli trials, the probabilities of k are
given by the binomial probability law:
pn k n
k
n−k
,
pk 1 − p
for k 0, 1, . . . , n,
1.8
Abstract and Applied Analysis
3
where pn k is the probability of k successes in n trials. For example, a communication system
transmits binary information over channel that introduces random bit errors with probability
ξ 10−3 . The transmitter transmits each information bit three times, an a decoder takes a
majority vote of the received bits to decide on what the transmitted bit was. The receiver
can correct a single error, but it will make the wrong decision if the channel introduces
two or more errors. If we view each transmission as a Bernoulli trial in which a “success”
corresponds to the introduction of an error, then the probability of two or more errors in
three Bernoulli trials is
pk ≥ 2 3
2
0.001 0.999 2
3
3
0.0013 ≈ 3 10−6 ,
1.9
see 9. Based on the q-integers Phillips introduced the q-analogue of well-known Bernstein
polynomials see 4, 5, 9, 11, 15. For f ∈ C0, 1, Phillips introduced the q-extension of
1.6 as follows:
Bn,q f | x
n
f
n
nq
n0
n
f
kq
kq
nq
n0
k
1 − xn−k
q
q
1.10
Bk,n x, q ,
for k, n ∈ Z
see 4, 5, 9, 11, 15. Here Bn,q f | x is called the q-Bernstein operator of order n for f. For
k, n ∈ Z , the q-Bernstein polynomial of degree n is defined by
Bk,n x, q n
k
xk 1 − xn−k
q ,
where x ∈ 0, 1.
1.11
q
Note that 1.11 is the q-extension of 1.7. That is, limq → 1 Bk,n x, q Bk,n x. For example,
B0,1 x, q 1 − x, B1,1 x, q x, and B0,2 x, q 1 − 2q x qx2 , . . .. Also Bk,n x, q 0 for k > n,
because nk q 0. For n, k ∈ Z , its probabilities are given by
px k n
k
xk 1 − xn−k
q ,
where x ∈ 0, 1.
1.12
q
This distributions are studied by several authors and they have applications in physics as
well as in approximation theory due to the q-Bernstein polynomials and the q-Bernstein
operators see 1–16. By the definition of the q-Bernstein polynomials, we easily see that
4
Abstract and Applied Analysis
the q-Bernstein basis is the probability mass function of q-binomial distribution. In this paper
we use the two q-analogues of exponential function as follows:
Eq x ∞
∞ n
xn
q 2 1−q x :q ∞ 1 1−q x q ,
nq !
n0
∞
xn
1
1
,
eq x ∞ nq !
1−q x :q ∞
1 1−q x q
n0
1.13
1.14
see 2–4, 6, 10. From 1.3, the improper q-integral is given by
∞/A
0
n
qn
q
f
fxdq x 1 − q
A
A
n∈Z
1.15
see 6, where the improper q-integral depends on A. The purpose of this paper is to give
some properties of several q-Bernstein type polynomials to express the q-integral on 0, 1 in
terms of q-beta and q-gamma functions. Finally, we derive some identities on the q-integral
of the product of several q-Bernstein type polynomials.
2. q-Integral Representation of q-Bernstein Polynomials
The gamma and beta functions are defined as the following definite integrals α > 0, β > 0:
Γα ∞
e−t tα−1 dt,
2.1
0
see 1–11, 14–16
B α, β 1
t
α−1
1 − t
β−1
dt ∞
0
0
tα−1
1 tαβ
dt.
2.2
From 2.1 and 2.2, we can derive the following equations:
Γα 1 αΓα,
ΓαΓ β
B α, β .
Γ αβ
2.3
As the q-extensions of 2.1 and 2.2, the q-gamma and q-beta functions are defined as the
following q-integrals α > 0, β > 0:
Γq α 1/1−q
0
xα−1 Eq −qx dq x
2.4
Abstract and Applied Analysis
5
see 2–6, 10,
Bq α, β 1
0
β−1
xα−1 1 − qx q dq x
2.5
see 2, 4, 6, 10.
By 2.4 and 2.5, we obtain the following lemma.
Lemma 2.1 see 2, 6. a Γq can be equivalently expressed as
Γq α 1−q
1−q
α−1
q
2.6
where α > 0.
α−1 ,
In particular, one has
Γq α 1 αq Γq α,
for α > 0, Γq 1 1.
2.7
b The q-gamma and q-beta functions are related to each other by the following two equations:
Γq αΓq β
Bq α, β ,
Γq α β
Bq α, ∞
Γq α α ,
1−q
where α > 0, β > 0.
2.8
Now one takes the q-integral for one q-Bernstein polynomial as follows: for n, k ∈ Z ,
q
−k
1
Bk,n
qx, q dq x 0
1
n
k
q
0
n−k
xk 1 − qx q dq x
n−k n − k
n k
l
q l0
l
q l0
−1 q
l1
2
1
xlk dq x
0
q
n−k n − k
n k
l
−1
n−k−l
q
n−k−l1
2
q
2.9
1
.
n − l 1q
Therefore, by 2.9, one obtains the following proposition.
Proposition 2.2. For n, k ∈ Z , one has
1
0
Bk,n qx, q dq x q
k
n−k n − k
n k
q l0
l
−1n−k−l q
q
n−k−l1
2
1
.
n − l 1q
2.10
6
Abstract and Applied Analysis
The Proposition 2.2 is closely related to the q-beta function which is given by
Bq n, m 1
0
Γq m m−1
xn−1 1 − qx q dq x,
1/1−q
xn−1 Eq −qx dq x,
2.11
2.12
0
see 2.5. From Lemma 2.1, one has
Bq n, m Γq mΓq n
,
Γq n m
where m, n ∈ N.
2.13
By 2.9 and 2.13, one gets
q
−k
1
Bk,n qx, q dq x n
0
k
Bq k 1, n − k 1
q
n Γq k 1Γq n − k 1
,
Γq n 2
k
2.14
where k > −1, n > k − 1.
q
Therefore, by 2.14, one obtains the following theorem.
Theorem 2.3. For n, k ∈ Z with k > −1 and n > k − 1, one has
1
Bk,n
qx, q dq x n
0
k
kq n − kq
q
Γq kΓq n − k
.
q − 1 kq 1
Γq n 2
2.15
By comparing the coefficients on the both sides of Proposition 2.2 and Theorem 2.3,
one obtains the following corollary.
Corollary 2.4. For n, k ∈ Z with k > −1 and n > k − 1, one has
n−k n − k
l0
l
q
n−k−l1
2
Γq k 1Γq n − k 1
q
.
−1n−k−l
Γq n 2
n − l 1q
2.16
Abstract and Applied Analysis
7
According to this result one can say that the q-integral of q-Bernstein polynomials from
0 to 1 is symmetric. Now one considers the q-integral for the multiplication of two q-Bernstein
polynomials which is given by the following relation:
1
0
Bk,n qx, q Bk,mqn−k1 x,qdq x
qnk−k2 2k
1
m
n
k
k
q
0
q
1
m
n
k
k
q
0
q
nm−2k
x2 k 1 − qx q
dq x
2.17
2k
unm−2k 1 − qu q dq u.
For n, k, m ∈ Z , one can derive the following equation 2.20 from 2.17:
1
0
Bk,n qx, q Bk,m q
n−k1
x, q dq x
qnk−k2 2k
2k l
2k
m n
l q −1 q
k
k
q
q l0
k
q
q l0
n m l − 2k 1q
2k−l
2k 2k
q
n
m l q −1
k
l1
2
2k−l1
2
2.18
.
n m − l 1q
Therefore, one obtains the following theorem.
Theorem 2.5. For m, n, k ∈ Z , one has
1
Bk,n qx, q Bk,m qn−k1 x, q dq x q
2k−l
2k 2k
q
n
m l q −1
nk−k2 2k
k
0
k
q
q l0
2k−l1
2
n m − l 1q
.
2.19
For m, n, k ∈ Z , by 2.5 and 2.9, one gets
1
0
Bk,n qx, q Bk,m qn−k1 x, q dq x
qnk−k2 2k
n
m
k
q
k
Bq n m − 2k 1, 2k 1.
2.20
q
Therefore, by Theorem 2.5 and 2.20, one obtains the following corollary.
Corollary 2.6. For k > −1 and n m − 2k > −1, one has
2k
l0
2k l
−12k−l q
q
2k−l1
2
n m − l 1q
Γq n m − 2k 1Γq 2k 1
.
Γq n m 2
2.21
8
Abstract and Applied Analysis
By the same method, the multiplication of three q-Bernstein polynomials is given by
the following relation: for k, n, m, s ∈ Z ,
1
0
Bk,n qx, q Bk,m qn−k1 x, q Bk,s qnm−2k1 x, q dq x
q3k2nk−3k2 mk
1
m
s
n
nms−3k
x3k 1 − qx q
dq x
k q k q k q 0
1
m
s
n
k
q
k
q
k
3k
unms−3k 1 − qu q dq u
0
q
3k
m
s 3k
n
k
q
k
q
k
l
q l0
q
q
k
q
k
l
q l0
−1l
1
unms−3kl dq u
0
q
3k
m
s 3k
n
k
l1
2
2.22
q
3k−l1
2
−1l3k
q
1
.
n m s − l 1q
Therefore, by 2.22, one obtains the following theorem.
Theorem 2.7. For n, m, s, k ∈ Z , one has
1
Bk,n qx, q Bk,m qn−k1 x, q Bk,s qnm−2k1 x, q dq x
0
q
3k2nk−3k2 mk
3k
n
m
s 3k
k
q
k
q
k
q l0
l
q
3k−l1
2
q
−1l3k
.
n m s − l 1q
2.23
From 2.5 and 2.22, one has
1
0
Bk,n qx, q Bk,m qn−k1 x, q Bk,s qnm−2k1 x, q dq x
q3k2nk−3k2 mk
m
s
n
Bq n m s − 3k 1, 3k 1.
k q k q k q
2.24
Therefore, by Theorem 2.7 and 2.24, one obtains the following corollary.
Corollary 2.8. For k > −1/3 and n m s − 3k > −1, one has
3k
3k
k0
l
q
3k−l1
Γq n m s − 3k 1Γq 3k 1
−1l3k q 2
.
Γq n m s 2
n m s − l 1q
2.25
Abstract and Applied Analysis
9
For s ∈ N, let n1 , n2 , . . . , ns , k ∈ Z . Then one has
1
0
i
s−1
l1 nl −ik1 x, q d x
q
Bk,n1 qx, q
B
k,n
q
i1
i1
s−1
2 s
qskk i1 ins−i −k 2 1
n1
n2
ns
n ···ns −sk
···
xsk 1 − qx q 1
dq x
k q k q
k q 0
n1
n2
k
k
q
···
k
q
k
q
n2
n1
k
sk
sk
ns ···
l
q l0
−1l q
k
l
q l0
1
q
2.26
xn1 ···ns −skl dq x
0
q
sk
sk
ns q
l1
2
sk−l1
−1lsk q 2
.
n1 · · · ns − l 1q
Therefore, by 2.26, one obtains the following theorem.
Theorem 2.9. For s ∈ N, let n1 , n2 , . . . , ns , k ∈ Z . Then one has
1
Bk,n1
s−1
i
nl −ik1
l1
Bk,ni1 q
x, q
dq x
qx, q
0
i1
s−1
s
qskk i1 ins−i −k 2 2
n1
k
···
q
lsk
sk sk
q
ns l q −1
k
q l0
sk−l1
2
2.27
n1 · · · ns − l 1q
.
By 2.5 and 2.26, we get
1
0
i
s−1
l1 nl −ik1 x, q d x
q
Bk,n1 qx, q
B
k,n
q
i1
i1
s−1
2 s
qskk i1 ins−i −k 2 n1
n2
ns
···
Bq sk 1, n1 · · · ns − sk 1
k q k q
k q
n2
n1
k
q
k
q
2.28
ns Γq sk 1Γq n1 · · · ns − sk 1
.
···
Γq n1 · · · ns 2
k
q
By comparing the coefficients on the both sides of Theorem 2.9 and 2.28, one obtains the
following corollary.
Corollary 2.10. For s ∈ N, let k > −1/s and n1 · · · ns − sk > −1. Then one has
sk
l0
sk l
−1lsk q
q
sk−l1
2
n1 · · · ns − l 1q
Γq sk 1Γq n1 · · · ns − sk 1
.
Γq n1 · · · ns 2
2.29
10
Abstract and Applied Analysis
For n ∈ Z , one gets
1
0
n
nl− l 1
2
B0,n qx, q
B
x,
q
dq x
q
l,n
l1
q
n
l1 nl−
l 1l
2
⎛ ⎞
1 n
n
n1 n1
2
⎠
⎝
2
x
dq x
1 − qx q
i q
0
i0
⎛ ⎞ n
n
n1
n1
⎠
⎝
Bq
1,
1
i q
2
2
i0
2.30
⎛ n1 ⎞ 2 Γq n 1
Γq nn 1/2 1
.
⎝ 2 ⎠
n
Γq nn 1 2
Γ
1
i
q
i1
Therefore, by 2.30, one obtains the following theorem.
Theorem 2.11. For n ∈ Z , one has
1
B0,n
n
nl− l 1
2
dq x
Bl,n q
x, q
qx, q
0
l1
q
⎛ l
l1 nl− 2 1l ⎝
n
n1 ⎞ 2 Γq n 1
Γq nn 1/2 1
⎠
.
2
n
Γq nn 1 2
i1 Γq i 1
2.31
From 2.30, one can also derive the following equation:
1
0
n
nl− l 1
2
q
B0,n qx, q
B
x,
q
dq x
l,n
l1
q
n
l1 nl−
l 1l
2
⎞
⎛
⎞n1 n1
1 n
2
n
n1 l
l1
⎜
⎟
l
⎝
⎠
2
x 2
dq x
⎝
⎠ −1 q
2
i q l0
0
i0
l
q
⎛
⎞
⎛
⎞n1 n1
n
2
n
l1
1
⎜
⎟
l
⎝
⎠
2
.
⎝
⎠ −1 q
2
1/2
l 1q
nn
i q l0
i0
l
q
⎛
2.32
Abstract and Applied Analysis
11
By comparing the coefficients on the both sides of Theorem 2.11 and 2.30, one can see that
nn1/2
nn1/2
l
q
−1l q
l1
2
nn 1/2 l 1q
l0
Bq
nn 1
nn 1
1,
1 .
2
2
2.33
Therefore, by 2.33, one obtains the following corollary.
Corollary 2.12. For n ∈ Z , one has
nn1/2
l0
nn1/2
l
q
−1l q
l1
2
nn 1/2 l 1q
Γq nn 1/2 1
Γq nn 1 2
2 .
2.34
References
1 S. Araci, D. Erdal, and J. Seo, “A study on the fermionic p-adic q-integral representation on Zp
associated with weighted q-Bernstein and q-Genocchi polynomials,” Abstract and Applied Analysis,
vol. 2011, Article ID 649248, 8 pages, 2011.
2 H. Exton, q-Hypergeometric Functions and Applications, Ellis Horwood Series: Mathematics and its
Applications, Ellis Horwood Ltd., Chichester, UK, 1983.
3 F. H. Jackson, “On q-definite integrals,” The Quarterly Journal of Pure and Applied Mathematics, vol. 41,
pp. 193–2036, 1910.
4 V. Gupta, T. Kim, J. Choi, and Y.-H. Kim, “Generating function for q-Bernstein, q-Meyer-König-Zeller
and q-Beta basis,” Automation Computers Applied Mathematics, vol. 19, pp. 7–11, 2010.
5 T. Kim, “A note on q-Bernstein polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 1,
pp. 73–82, 2011.
6 S.-H. Rim, J.-H. Jin, E.-J. Moon, and S.-J. Lee, “Some identities on the q-Genocchi polynomials of
higher-order and q-Stirling numbers by the fermionic p-adic integral on Zp ,” International Journal of
Mathematics and Mathematical Sciences, vol. 2010, Article ID 860280, 14 pages, 2010.
7 N. K. Govil and V. Gupta, “Convergence of q-Meyer-König-Zeller-Durrmeyer operators,” Advanced
Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 97–108, 2009.
8 T. Kim, “q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,”
Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008.
9 T. Kim, “q-polynomials, qstirling numbers and q-Bernoulli polynomials,” http://arxiv.org/abs/
1008.4547.
10 T. Kim, “q-extension of the Euler formula and trigonometric functions,” Russian Journal of Mathematical Physics, vol. 14, no. 3, pp. 275–278, 2007.
11 B. A. Kupershmidt, “Reflection symmetries of a note on q-Bernstein polynomials,” Journal of Nonlinear
Mathematical Physics, vol. 12, pp. 412–422, 2005.
12 A. D. Sole and V. G. Kac, “On integral representations of q-Gamma and q-beta functions,”
http://arxiv.org/abs/math/0302032.
13 L.-C. Jang, “A family of Barnes-type multiple twisted q-Euler numbers and polynomials related to
Fermionic p-adic invariant integrals on Zp ,” Journal of Computational Analysis and Applications, vol. 13,
no. 2, pp. 376–387, 2011.
14 T. Kim, “Some formulae for the q-Bernstein polynomials and q-deformed binomial distributions,”
Journal of Computational Analysis and Applications. In press.
15 G. M. Phillips, “Bernstein polynomials based on the q-integers,” Annals of Numerical Mathematics, vol.
4, no. 1-4, pp. 511–518, 1997.
16 T. Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the
fermionic p-adic integral on Zp ,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491,
2009.
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Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Optimization
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014