Title

q-Analogues of Freud weights and nonlinear difference equations

Authors

Authors

M. E. H. Ismail;Z. S. I. Mansour

Comments

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Abbreviated Journal Title

Adv. Appl. Math.

Keywords

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

Abstract

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.

Journal Title

Advances in Applied Mathematics

Volume

45

Issue/Number

4

Publication Date

1-1-2010

Document Type

Article

Language

English

First Page

518

Last Page

547

WOS Identifier

WOS:000282113100005

ISSN

0196-8858

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