Zeros of a certain class of Gauss hypergeometric polynomials

TL;DR

This paper investigates the asymptotic distribution of zeros of a class of Gauss hypergeometric polynomials with complex parameters, showing they cluster on specific curves as the degree increases, extending previous conjectures and results.

Contribution

It provides a partial proof of a conjecture on zero distribution for hypergeometric polynomials with complex parameters, generalizing earlier work to include complex b5.

Findings

Zeros cluster on a specific curve as degree tends to infinity

Generalizes previous results to complex parameter b5

Supports conjecture on zero distribution of hypergeometric polynomials

Abstract

In this paper, we give results that partially prove a conjecture which was discussed in our previous work (arXiv:1307.4991). More precisely, we prove that as $n\to \infty,$ the zeros of the polynomial$${}_{2}\text{F}_{1}\left[ \begin{array}{c} -n, \alpha n+1\\ \alpha n+2 \end{array} ; \begin{array}{cc} z \end{array}\right]$$ cluster on a certain curve defined as a part of a level curve of an explicit harmonic function. This generalizes work by Boggs, Driver, Duren et. al, to a complex parameter $\alpha$.