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 2 Discrete Dynamics in Nature and Society 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 4 Discrete Dynamics in Nature and Society 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 5 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 6 Discrete Dynamics in Nature and Society 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 7 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 8 Discrete Dynamics in Nature and Society 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 9 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 10 Discrete Dynamics in Nature and Society 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 11 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. References 1 S. Bernstein, “Demonstration du theoreme de Weierstrass, fondee sur le calcul des probabilities,” Communications of the Kharkov Mathematical Society, vol. 2, no. 13, pp. 1–2, 1912-1913. 2 M. Acikgoz and S. Araci, “A study on the integral of the product of several type Bernstein polynomials,” IST Transaction of Applied Mathematics-Modelling and Simulation. In press. 12 Discrete Dynamics in Nature and Society 3 N. K. Govil and V. Gupta, “Convergence of q-Meyer-König-Zeller-Durrmeyer operators,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 181–189, 2009. 4 M. Acikgoz and S. 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