Mixed recurrence relations and interlacing of the zeros of some \(q\)-orthogonal polynomials from different sequences
type: Item
instance of: scholarly article
title: Mixed recurrence relations and interlacing of the zeros of some \(q\)-orthogonal polynomials from different sequences
zbMATH Open document ID: 1224.26054
author: K. Jordaan
author: F. Toókos
published in: Acta Mathematica Hungarica
publication date: 2011-04-26T00:00:00Z
review text: The fact that the zeros of orthogonal polynomials of degree \(n\) and \(n-1\) interlace has been studied extensively. For \(n\in\mathbb{N}\) let us denote by \(p_n\) an orthogonal polynomial of degree \(n\) on the interval \((c,d)\). The author presents two generalizations of known results. Let \(f(x)=a_1(x) p_n(x) +a_0(x)p_{n-1}(x)\) be continuous. (I) If \(a_0\) and \(a_1\) are continuous and of constant sign on \((c,d)\), then \(f\) has at least one zero between any two consecutive zeros of \(p_{n-i}\). (Cf. [\textit{C. Brezinski}, \textit{K. A. Driver} and \textit{M. Redivo-Zaglia}, Appl. Numer. Math. 48, No. 2, 157--168 (2004; Zbl 1047.33002)].) (II). If \(f\) is a polynomial and both \(a_0\) and \(a_1\) are continuous and of constant sign on \((c,d)\), then all zeros of \(f\) are real and simple; the zeros of \(f\) and \(p_n\) interlace and those of \(f\) and \(p_{n-1}\) interlace. (Cf. [\textit{T. S. Chihara}, An introduction to orthogonal polynomials. Mathematics and its Applications. Vol. 13. New York: Gordon and Breach, Science Publishers (1978; Zbl 0389.33008), Exercise 5.4].) With the aid of these tools, the author gives interlacing properties in the following cases: 1. Al-Salam-Chihara polynomials with different parameters; 2. \(q\)-Meixner-Pollaczek polynomials with different parameters; 3. continuous \(q\)-ultraspherical polynomials with different parameters; 4. \(q\)-Laguerre polynomials with different parameters.
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 26C10
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 5881201
zbMATH Keywords: \(q\)-orthogonal polynomial
zbMATH Keywords: interlacing of zeros
zbMATH Keywords: separation of zeros
zbMATH Keywords: Al-Salam-Chihara
zbMATH Keywords: continuous \(q\)-ultraspherical
zbMATH Keywords: \(q\)-Meixner-Pollaczek
zbMATH Keywords: \(q\)-Laguerre
MaRDI profile type: Publication
full work available at URL: s10474-010-9176-9
OpenAlex ID: W2136236993
cites work: Conjecture on the interlacing of zeros in complex Sturm-Liouville problems
cites work: Quasi-orthogonality with applications to some families of classical orthogonal polynomials.
cites work: Q4170993
cites work: Convergence of Extended Lagrange Interpolation
cites work: On the behaviour of zeros of Jacobi polynomials
cites work: Interlacing of the zeros of Jacobi polynomials with different parameters
cites work: Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials
cites work: Q4040884
cites work: Bericht über die Nullstellen der Laguerreschen und der Hermiteschen Polynome
cites work: Über die Nullstellen der Hermiteschen Polynome
cites work: Interlacing theorems for the zeros of some orthogonal polynomials from different sequences
cites work: Q4284666
cites work: Interlacing properties of the zeros of the orthogonal polynomials and approximation of the Hilbert transform
cites work: The q-analogue of the Laguerre polynomials
cites work: Q4387420
cites work: Linear Combinations of Orthogonal Polynomials Generating Positive Quadrature Formulas
cites work: The Zeros of Special Functions from a Fixed Point Method
cites work: Interlacing of the zeros of contiguous hypergeometric functions
cites work: Inequalities for the Zeros of Legendre Polynomials and Related Functions
DOI: 10.1007/S10474-010-9176-9
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description: scientific article; zbMATH DE number 5881201