q-Analogues of Freud weights and nonlinear difference equations

    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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