Interlacing of zeros of orthogonal polynomials under modification of the measure

    Authors

    D. K. Dimitrov; M. E. H. Ismail;F. R. Rafaeli

    Comments

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    Abbreviated Journal Title

    J. Approx. Theory

    Keywords

    Orthogonal polynomials; Classical orthogonal polynomials; q-orthogonal; polynomials; Zeros; Interlacing; Monotonicity; LINEAR-COMBINATIONS; DIFFERENT SEQUENCES; JACOBI-POLYNOMIALS; Mathematics

    Abstract

    We investigate the mutual location of the zeros of two families of orthogonal polynomials. One of the families is orthogonal with respect to the measure d mu(x), supported on the interval (a, b) and the other with respect to the measure vertical bar x - c vertical bar(tau)vertical bar x - d vertical bar(gamma) d mu(x), where c and d are outside (a, b). We prove that the zeros of these polynomials, if they are of equal or consecutive degrees, interlace when either 0 < tau, gamma < = 1 or gamma = 0 and 0 < tau < = 2. This result is inspired by an open question of Richard Askey and it generalizes recent results on some families of orthogonal polynomials. Moreover, we obtain further statements on interlacing of zeros of specific orthogonal polynomials, such as the Askey-Wilson ones. (c) 2013 Elsevier Inc. All rights reserved.

    Journal Title

    Journal of Approximation Theory

    Volume

    175

    Publication Date

    1-1-2013

    Document Type

    Article

    Language

    English

    First Page

    64

    Last Page

    76

    WOS Identifier

    WOS:000325121000004

    ISSN

    0021-9045

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