Interlacing of zeros of orthogonal polynomials under modification of the measure

Author(s)

Dimitrov, Dimitar K. ^[1] ; Ismail, Mourad E. H. ^{[2, 3]} ; Rafaeli, Fernando R. ^[1]

Affiliation

^[1] Univ Estadual Paulista UNESP, Inst Biociencias Letras & Ciencias Exatas, Dept Matemat Aplicada, Jaboticabal - Brazil

^[2] Univ Cent Florida, Dept Math, Orlando, FL 32816 - USA

^[3] King Saud Univ, Coll Sci, Dept Math, Riyadh 11451 - Saudi Arabia

• Document type: Journal article
• Source: Journal of Approximation Theory; v. 175, p. 64-76, NOV 2013.
• Web of Science Citations: 2

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. (AU)

• FAPESP's process: 09/13832-9 - Orthogonal polynomials, special functions and applications
• Grantee: Dimitar Kolev Dimitrov
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 11/00658-0 - Zeros of orthogonal polynomials
• Grantee: Fernando Rodrigo Rafaeli
• Support Opportunities: Research Grants - Young Investigators Grants