Properties of convergence for the q-Meyer-König and Zeller operators ✩

In this paper, we discuss properties of convergence for the q-Meyer-König and Zeller operators Mn,q . Based on an explicit expression for Mn,q (t2, x) in terms of q-hypergeometric series, we show that for qn ∈ (0, 1], the sequence (Mn,qn(f ))n 1 converges to f uniformly on [0, 1] for each f ∈ C[0, 1] if and only if limn→∞qn = 1. For fixed q ∈ (0, 1), we prove that the sequence (Mn,q (f )) converges for each f ∈ C[0, 1] and obtain the estimates for the rate of convergence of (Mn,q (f )) by the modulus of continuity of f , and the estimates are sharp in the sense of order for Lipschitz continuous functions. We also give explicit formulas of Voronovskaya type for the q-Meyer-König and Zeller operators for fixed 0