Q-Analogues of Freud weights and nonlinear difference equations

Abstract

In this paper we derive the nonlinear recurrence relation for the recursion coefficients β n of polynomials orthogonal with respect to q-analogues of Freud exponential weights. An asymptotic relation for β 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 β n 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.