Zeros of the hypergeometric polynomial F(-n,b;c;z)

TL;DR

This paper reviews recent findings on the zero distribution of Gauss hypergeometric polynomials with arbitrary parameters, linking their zeros to those of Jacobi polynomials, and extends understanding beyond previously analyzed cases.

Contribution

It provides a comprehensive review of zero behaviors for hypergeometric polynomials with arbitrary parameters, connecting these to Jacobi polynomial results and broadening the scope of analysis.

Findings

Number of real zeros can be determined for all real parameters.

Intervals of zeros are identified based on parameter values.

Results extend known cases to more general parameter settings.

Abstract

Our interest lies in describing the zero behaviour of Gauss hypergeometric polynomials $F(-n,b; c; z)$ where $b$ and $c$ are arbitrary parameters. In general, this problem has not been solved and even when $b$ and $c$ are both real, the only cases that have been fully analyzed impose additional restrictions on $b$ and $c$. We review recent results that have been proved for the zeros of several classes of hypergeometric polynomials $F(-n,b; c; z)$ where $b$ and $c$ are real. We show that the number of real zeros of $F(-n,b; c; z)$ for arbitrary real values of the parameters $b$ and $c$, as well as the intervals in which these zeros (if any) lie, can be deduced from corresponding results for Jacobi polynomials.