Interlacing of zeros of orthogonal polynomials under modification of the measure

JOURNAL OF APPROXIMATION THEORY

卷:175

Article

Dimitrov, Dimitar K.¹  Ismail, Mourad E. H.^2,3  Rafaeli, Fernando R.¹ 

[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

关键词: Orthogonal polynomials;    Classical orthogonal polynomials;    q-orthogonal polynomials;    Zeros;    Interlacing;    Monotonicity;   

DOI  :  10.1016/j.jat.2013.07.007

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

Free