q-Analogues of Freud weights and nonlinear difference equations
Abbreviated Journal Title
Adv. Appl. Math.
Orthogonal polynomials; Nonlinear difference equations; q-Analogue of; Freud weights and Freud's equations; Discrete Painleve property; Plancherel-Rotach type asymptotics; Bernstein's approximation problem; DISCRETE PAINLEVE EQUATIONS; Q-ORTHOGONAL POLYNOMIALS; EXPONENTIAL; WEIGHTS; LADDER OPERATORS; GREATEST ZERO; COEFFICIENTS; RECURRENCE; Mathematics, Applied
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.
Advances in Applied Mathematics
"q-Analogues of Freud weights and nonlinear difference equations" (2010). Faculty Bibliography 2010s. 291.