On the zeros of a class of polynomials defined by a three term recurrence relation
type: Item
instance of: scholarly article
title: On the zeros of a class of polynomials defined by a three term recurrence relation
zbMATH Open document ID: 0801.12001
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-11-29T00:00:00Z
review text: The polynomial \(P_{N+1} (x)\) is defined recursively by \(P_ 0(x)=0\), \(P_ 1(x) =1\), and \(\alpha_ n P_{n+1} (x)+ \alpha_{n-1} P_{n-1} (x)+ b_ n P_ n(x)=x d_ n P_ n(x)\) \((1\leq n\leq N)\); under suitable hypotheses on the coefficient sequences, it has real and simple zeros. This generalizes a result of Van Doorn and is equivalent to one of K. I. Sato. The author's proof uses the fact that the zeros of the polynomials are the eigenvalues for the problem \((AV^*+ VA+ B)f= \lambda Df\) (\(f\neq 0\)), where \(f\) is in an \(N\)-dimensional Hilbert space, \(V\) is a shift and \(A\), \(B\), \(D\) are diagonal operators.
Mathematics Subject Classification ID: 12D10
Mathematics Subject Classification ID: 26C10
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 584060
zbMATH Keywords: zeros of polynomials
zbMATH Keywords: finite dimensional Hilbert space
zbMATH Keywords: eigenvalues
author: E. K. Ifantis
author: P. N. Panagopoulos
reviewed by: Edward J. Barbeau
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1091
OpenAlex ID: W2094466162
DOI: 10.1006/JMAA.1994.1091
description: scientific article; zbMATH DE number 584060