q-Analogues of Freud weights and nonlinear difference equations

In this paper we derive the nonlinear recurrence relation for the recursion coefficients beta(n) of polynomials orthogonal with respect to q-analogues of Freud exponential weights. An asymptotic relation for beta(n) is given under assuming a certain smoothing condition and the Plancherel-Rotach asymptotic for the corresponding orthogonal polynomials is derived. Special interest is paid to the case m = 2. We prove that the nonlinear recurrence relation of beta(n) in this case obeys the discrete Painleve property. Motivated by Lew and Quarles, we study possible periodic solutions for a class of nonlinear difference equations of second order. Finally we prove that the Bernstein approximation problem associated to the weights under consideration has a positive solution. (C) 2010 Elsevier Inc. All rights reserved