On the limit distributions of the zeros of Jonquière polynomials and generalized classical orthogonal polynomials
type: Item
instance of: scholarly article
title: On the limit distributions of the zeros of Jonquière polynomials and generalized classical orthogonal polynomials
zbMATH Open document ID: 0832.33004
author: Jutta Faldey
author: Wolfgang Gawronski
published in: Journal of Approximation Theory
publication date: 1995-05-18T00:00:00Z
review text: Some sequences of polynomials with real zeros are investigated, with emphasis on the asymptotic distribution of the zeros. First the Jonquière polynomials \(J_k (z)\) are investigated, which are defined by \[ {{J_k (z)} \over {(1- z)^{k+ 1}}}= \sum_{n=0}^\infty n^k z^n. \] The zero distribution was found earlier by Gawronski and Stadtmüller, but now the authors consider a generalization of the form \[ {{J_k (z)} \over {\prod_{j= 1}^{k+ 1} (1- ze^{\beta_{k+ 1,j}})}}= \sum_{n= 0}^\infty E_k (n) z^n, \] where the \(E_k\) are exponential polynomials satisfying \(Q_{k+ 1} (d/dx) E_k (x) =0\), with \(Q_{k+1}\) a polynomial of degree \(k+1\) with zeros \(\beta_{k+ 1,j}\), \(j=1, 2, \dots, k+1\). If the asymptotic distribution \(\mu\) of the zeros of \(Q_{k+1}\) for \(k\to \infty\) is known, then the authors give the asymptotic distribution of the zeros of \(J_k\) in terms of \(\mu\). In the second part of the paper, the authors give the asymptotic zero distribution of polynomials which are a solution of a second order differential equation for which the coefficients have prescribed asymptotic behaviour as the degree of the polynomials tends to infinity. This case covers Jacobi polynomials \(P_n^{(\alpha_n, \beta_n)}\) where the parameters may depend on the degree \(n\). Also Laguerre polynomials \(L_n^{( \alpha_n)}\) and generalized Hermite polynomials \(H_n^{( \alpha_n)}\), with weight function \(|x|^{2 \alpha_n} e^{- x^2}\) on \((- \infty, \infty)\), are considered.
reviewed by: Walter Van Assche
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 756617
zbMATH Keywords: classical orthogonal polynomials
zbMATH Keywords: Jonquière polynomials
zbMATH Keywords: zero distribution
MaRDI profile type: Publication
full work available at URL: jath.1995.1047
OpenAlex ID: W2084325476
DOI: 10.1006/JATH.1995.1047
description: scientific article; zbMATH DE number 756617