Some Identities on the q-Integral Representation of the Product of Several q-Bernstein-Type Polynomials

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 p k ≥ 2 ( 3 2 ) 0.001 2 0.999 ( 3 3 ) 0.001 3 ≈ 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 ∈ C 0, 1 , Phillips introduced the q-extension of 1.6 as follows: Bn,q ( f | x n ∑ n 0 f ( k q n q )( n k )


Introduction
Let q ∈ R with 0 ≤ q < 1.We assume that q-number is defined by x q 1 − q x / 1 − q and 0 q 0. Note that lim q → 1 x q x.The q-derivative of a map f : R → R at x ∈ R \ {0} is given by For n ∈ N, by 1.1 , we get D n q x n n q n−1 q • • • 2 q 1 q n 1 !.The q-binomial formula is given by a b n q n−1 i 0 a bq i n l 0 n l q q l 2 a n−l b l 1.2 see 2, 5, 7-11 , where n k q n q !/ k q !n − k q !n q n − 1 q • • • n − k 1 q / k q !.

Abstract and Applied Analysis
For a, b ∈ R, the Jackson q-integral of f : R → R is defined by b a f x d q x 1 − q ∞ n 0 q n bf bq n − af aq n 1.3 see 1, 2, 5, 6, 9, 12, 13 .From 1.2 , we note that By 1.2 and 1.4 , we get

1.5
Let C 0, 1 denote the set of continuous function on 0, 1 .For f ∈ C 0, 1 , Bernstein introduced the following well-known linear operators see 1, 4, 9, 11, 14 : Here B n 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 3,4,[11][12][13][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: where p n 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 p k ≥ 2 3 2 0.001 2 0.999 3 3 0.001 3 ≈ 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 ∈ C 0, 1 , Phillips introduced the q-extension of 1.6 as follows: 1.10 see 4, 5, 9, 11, 15 .Here B n,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 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 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: x n n q !, 1.14 see 2-4, 6, 10 .From 1.3 , the improper q-integral is given by 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.

q-Integral Representation of q-Bernstein Polynomials
The gamma and beta functions are defined as the following definite integrals α > 0, β > 0 : From 2.1 and 2.2 , we can derive the following equations: 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 : see 2, 4, 6, 10 .By 2.4 and 2.5 , we obtain the following lemma.

2.6
In particular, one has b The q-gamma and q-beta functions are related to each other by the following two equations: , where α > 0, β > 0. 2.8 Now one takes the q-integral for one q-Bernstein polynomial as follows: for n, k ∈ Z , 2.9 Therefore, by 2.9 , one obtains the following proposition.
Proposition 2.2.For n, k ∈ Z , one has Abstract and Applied Analysis The Proposition 2.2 is closely related to the q-beta function which is given by 1/ 1−q 0 x n−1 E q −qx d q x, 2.12 see 2.5 .From Lemma 2.1, one has , where m, n ∈ N.

2.13
By 2.9 and 2.13 , one gets

2.14
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 By comparing the coefficients on the both sides of Proposition 2.2 and Theorem 2.3, one obtains the following corollary.

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:

2.17
For n, k, m ∈ Z , one can derive the following equation 2.20 from 2.17 :

2.18
Therefore, one obtains the following theorem.

2.19
For m, n, k ∈ Z , by 2.5 and 2.9 , one gets

2.20
Therefore, by Theorem 2.5 and 2.20 , one obtains the following corollary.

2.21
By the same method, the multiplication of three q-Bernstein polynomials is given by the following relation: for k, n, m, s ∈ Z ,

2.22
Therefore, by 2.22 , one obtains the following theorem.

2.23
From 2.5 and 2.22 , one has

2.24
Therefore, by Theorem 2.7 and 2.24 , one obtains the following corollary.n m s − l 1 q Γ q n m s − 3k 1 Γ q 3k 1 Γ q n m s 2 .

2.26
Therefore, by 2.26 , one obtains the following theorem.

2.28
By comparing the coefficients on the both sides of Theorem 2.9 and 2.28 , one obtains the following corollary.

2.29
For n ∈ Z , one gets

2.30
Therefore, by 2.30 , one obtains the following theorem.B l,n q nl− l 2 1 x, q d q x q n l 1 nl− l Γ q n n 1 2 .

2.31
From 2.30 , one can also derive the following equation: 1 n n 1 /2 l 1 q .

2.32
By comparing the coefficients on the both sides of Theorem 2.11 and 2.30 , one can see that n n 1 /2 l q −1 l q l 1 2 n n 1 /2 l 1 q B q n n 1 2 1, n n 1 2 1 .

2.33
Therefore, by 2.33 , one obtains the following corollary.