Pólya frequency sequences and real zeros of some \(_3F_2\) polynomials
type: Item
instance of: scholarly article
title: Pólya frequency sequences and real zeros of some \(_3F_2\) polynomials
zbMATH Open document ID: 1114.33006
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-06-06T00:00:00Z
review text: The authors prove that the zeros of three distinct classes of hypergeometric polynomials of \(_{3}F_{2}\) type are all real. More precisely, they show that the zeros of the polynomial \[ _{3}F_{2}\left(\begin{aligned} -n,\,-m,\,a-1/2 a,\,2a-1 \end{aligned} ;\,x \right),\quad a>0, \;m,\,n \in \mathbb{N}, \] are all real and negative. For \(n=m\) and \(a=1\) this implies that all zeros of the polynomial \[ _{3}F_{2}\left(\begin{aligned} -n,\,-n,\,1/2 1,\,1 \end{aligned} ;\,4x \right)=\sum_{k=0}^{n}\Big({n \atop k}\Big)^{2}\,\Big({2k \atop k}\Big)\,x^{k} \] are real and negative which settles a conjecture of C. Greene and H. Wilf. The same result is obtained for two other families of \(_{3}F_{2}\) hypergeometric polynomials. These results have connection with the theory of Pólya frequency sequences and functions. The authors also derive the asymptotic distribution (as \(n\rightarrow \infty\)) of the zeros of two of the classes of polynomials considered in the paper.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 26C10
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 12D10
zbMATH DE Number: 5161755
zbMATH Keywords: \(_{3}F_{2}\) polynomials
zbMATH Keywords: hypergeometric polynomials
zbMATH Keywords: Pólya frequency sequences
zbMATH Keywords: Pólya frequency function
zbMATH Keywords: total positivity
author: Kathy Driver
author: K. Jordaan
author: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.10.080
OpenAlex ID: W1991155617
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DOI: 10.1016/J.JMAA.2006.10.080
description: scientific article; zbMATH DE number 5161755