Zeros of a certain class of Gauss 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 $\alpha$.