A class of hypergeometric polynomials with zeros on the unit circle: extremal and orthogonal properties and quadrature formulas
type: Item
instance of: scholarly article
title: A class of hypergeometric polynomials with zeros on the unit circle: extremal and orthogonal properties and quadrature formulas
zbMATH Open document ID: 1268.33007
author: Dimitar K. Dimitrov
author: Mourad E. H. Ismail
author: A. Sri Ranga
published in: Applied Numerical Mathematics
publication date: 2013-02-21T00:00:00Z
review text: The theory of special functions has been increasingly applied to the study of polynomials. Special hypergeometric functions are of great interest for the study of orthogonal polynomials on the real line. Consider the hypergeometric function \(_2F_1(a,b;c;z)\), commonly referred to as Gauss' hypergeometric function, i.e., \[ _2F_1 (a,b;c;z)=\sum_{n=0}^{\infty}{\frac{(a)_{n}(b)_{n}}{(c)_{n}} \frac{z^n}{n!}}, \] where \((\cdot)_{n}\) denotes the Pochhammer symbol and \(c\neq0,-1,-2,\dots\) In the cases where any of the parameters \(a\) and \(b\) are negative integers \(-m\), the series expansion terminates at \(n=m\), which reduces the function \(_2F_1(a,b;c;z)\) to a polynomial of degree \(m\). In the paper, the authors discuss some properties of two classes of hypergeometric polynomials, namely \[ R_{m}(b;z)=_2F_1 (-m,b;b+ \bar{b};1-z), \quad m\geq 0, \] \[ S_{m}(b;z)=_2F_1 (-m,b+1;b+ \bar{b}+1;1-z), \quad m\geq 0, \] where \(\operatorname{Re}(b)>0\). The class \(R_{m}(b;z)\) is of interest, since, when these polynomials are considered as functions of \(\operatorname{Im}(b)\), then they are Meixner-Pollaczek polynomials. The other class \(S_{m}(b;z)\), on the other hand, turns out to be interesting from the point of view of the theory of orthogonal polynomials on the unit circle. Also, the authors point out that \(\{R_{m}(b;z)\}\) is a special sequence of para-orthogonal polynomials associated with the sequence \(\{S_{m}(b;z)\}\) of orthogonal polynomials in the unit circle, or Szegő polynomials. The authors derive an interpolatory quadrature formula on the zeros of \(R_{m}(b;z)\). Also, related quadrature formulas for the interval \((-1,1)\) are derived which is based on the zeros of the auxiliary function \(G_{m}(\lambda,\eta;\chi)\). The latter can be defined as \[ G_{m}(\lambda,\eta;\chi)=\frac{(2\lambda)_{m}}{(\lambda)_{m}}(4z)^{-m/2}R_{m}(b;z), \quad m\geq 0, \] where \(\lambda\), \(\eta\) and \(\chi\) are such that \(b=\lambda +i\eta\) and \(2\chi=z^{1/2}+z^{-…
reviewed by: Luan Carlos de S. M. Ozelim
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 65D32
zbMATH DE Number: 6138080
zbMATH Keywords: hypergeometric polynomials
zbMATH Keywords: para-orthogonal polynomials
zbMATH Keywords: quadrature rules
MaRDI profile type: Publication
full work available at URL: j.apnum.2012.11.002
OpenAlex ID: W2012354376
DOI: 10.1016/J.APNUM.2012.11.002
description: scientific article; zbMATH DE number 6138080