Some approximation properties of q-Chlodowsky operators Harun Karsli a , Vijay Gupta b, * a Ankara University, Faculty of Sciences, Department of Mathematics, 06100 Tandogan-Ankara, Turkey b School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi 110075, India Abstract In the present paper, we introduce the q-analog of Chlodowsky operators. We study some approximation properties of these new operators, which include the rate of convergence and investigation of the monotonocity property of q-Chlodow- sky operators. Ó 2007 Elsevier Inc. All rights reserved. Keywords: q-Chlodowsky polynomials; Rate of convergence; Modulus of continuity; Peetre-K functional; Lipschitz space 1. Introduction Phillips [4] introduced the generalization of Bernstein polynomials based on q-integers. Very recently Aral and Gupta [1] studied and established some approximation properties for q-Szasz Mirakyan operators. We now define the q analogue of Chlodowsky operators. Before introducing the operators, we mention certain properties of the q-calculus: For any fixed real number q > 0 and non-negative integer r the q-integer of the number r is defined by ½r q ¼ ð1  q r Þ=ð1  qÞ; q 6¼ 1; r; q ¼ 1:  The q-factorial is defined by ½r q ! ¼ ½r q ½r  1 q ... ½1 q ; r ¼ 1; 2; 3; ... ; 1; r ¼ 0:  and q-Binomial coefficient can be defined as n r  q ¼ ½n q ! ½r q !½n  r q ! ; for integers n P r P 0: 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.085 * Corresponding author. E-mail addresses: karsli@science.ankara.edu.tr (H. Karsli), vijaygupta2001@hotmail.com (V. Gupta). Available online at www.sciencedirect.com Applied Mathematics and Computation 195 (2008) 220–229 www.elsevier.com/locate/amc