q-Analogues of Freud weights and nonlinear difference equations

In this paper we derive the nonlinear recurrence relation for the recursion coefficients βn of polynomials orthogonal with respect to q-analogues of Freud exponential weights. An asymptotic relation for βn 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 βn in this case obeys the discrete Painlevé 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.