Some approximation properties of q-Durrmeyer operators Vijay Gupta School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi 110 075, India Abstract In the present paper, we introduce a simple q analogue of well known Durrmeyer operators. We first estimate moments of q-Durrmeyer operators. We also establish the rate of convergence for q-Durrmeyer operators. Ó 2007 Elsevier Inc. All rights reserved. Keywords: q-Factorial; q-Integers; q-Durrmeyer operators; q-Beta function; Modulus of continuity 1. Introduction To approximate Lebesgue integrable functions on the interval [0, 1] Durrmeyer introduced the integral modification of the well known Bernstein polynomials. In 1981 Derriennic [1] first studied these operators in details. The Durrmeyer operators are defined as D n ðf ; xÞ¼ðn þ 1Þ X n k¼0 p nk ðxÞ Z 1 0 f ðtÞp nk ðtÞ dt; ð1Þ where the Bernstein basis function is defined by p nk ðxÞ¼ n k  x k ð1  xÞ nk . In the last decade Phillips [8] intro- duced the q-analogue of Bernstein polynomials. After that several researchers have studied the q-analogue of Bernstein polynomials and established many interesting approximation properties. For important work in this direction, we refer the readers to [6,7,11–15] etc. As Durrmeyer operators approximate integrable functions on the interval [0, 1], this motivated us to introduce the q analogue of the Durrmeyer operators. Before introduc- ing the operators, we mention some basic definitions of q calculus. Let q > 0. For each nonnegative integer k, the q-integer [k] and the q-factorial [k]! are respectively defined by ½k :¼ ð1  q k Þ=ð1  qÞ; q 6¼ 1; k; q ¼ 1;  0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.07.056 E-mail address: vijaygupta2001@hotmail.com Available online at www.sciencedirect.com Applied Mathematics and Computation 197 (2008) 172–178 www.elsevier.com/locate/amc