Higher Order Generalization of q -Bernstein Operators Pembe Sabancıgil Eastern Mediterranean University Department of Mathematics Gazimagusa Mersin 10 Turkey Email: pembe.sabancigil@emu.edu.tr Abstract We introduce higher order generalization of the q-Bernstein operators. Then we study ap- proximation properties and a Voronovskaja-type theorem for higher order q-Bernstein operators. 1 Introduction Nowadays it is known that the theory of q-calculus plays an important role on analytic number theory and theoretical physics. For example, various applications of this theory have appeared in the study of hypergeometric series [1], in the approximation theory [2], [17], [18] while some other important applications have been related with the quantum theory. In this paper, using the moment estimates from [9] and with the techniques of the works [5], [6], we study the approximation properties of an rth order generalization of the q-Bernstein polynomials. We first recall some basic definitions used in the paper. The q-Bernstein operators are given by B n,q (f ; x)= n  k=0 f  [k] [n]  p n,k (q; x) , n ∈ N, 0 ≤ x ≤ 1, p n,k (q; x)=  n k  x k n−k−1  s=0 (1 − q s x) . Recall that [n]=[n] q and  n k  denotes the q-integers and q-Gaussian binomial, which are defined, respectively by [n]=  (1 − q n ) / (1 − q) , if q =1 n, if q =1 ,  n k  = [n]! [k]! [n − k]! , where [n]! denotes the q-factorial given by [n]! =  [n] ... [2] [1] , if n ≥ 1, 1, if n =0. After q-Bernstein polynomials were introduced by Phillips [15] they have been the object of several investigations in approximation theory (cf. [3]-[21]). Surveys of results on the q-Bernstein polynomials together with comprehensive lists of references on the subject are given in [12]. We introduce a new sequence of positive linear operators so-called higher (rth) order q-Bernstein operators. Definition 1 Let r ∈ N ∪{0} be a fixed number. For f ∈ C r [0, 1] and n ∈ N we define the rth order generalization of the q-Bernstein operators as follows B [r] n,q (f ; x) := n  k=0 p n,k (q; x) r  j=0 1 j ! f (j)  [k] [n]  x − [k] [n]  j . 1