Asymptotic distribution of zeros of a certain class of hypergeometric polynomials

TL;DR

This paper investigates the asymptotic distribution of zeros of a specific class of hypergeometric polynomials, revealing their clustering on algebraic curves and proposing a conjecture supported by experimental evidence.

Contribution

It describes the zero distribution of hypergeometric polynomials using algebraic curves and introduces a conjecture in a degenerate case, extending previous work.

Findings

Zeros cluster on level curves derived from algebraic equations

Identification of the zero distribution pattern as parameters tend to infinity

Proposed conjecture supported by experimental evidence

Abstract

We study the weak asymptotic behavior of the zeros of a family of a certain class of (generalized) hypergeometric polynomials, using the associated hypergeometric differential equation, as the parameters go to infinity. We describe the curve complex on which the zeros cluster, as level curves associated to integrals on an algebraic curve derived from the equation. In a certain degenerate case we make a precise conjecture, based and generalizing work by Duren, Driver et. al, and present experimental evidence for it.