Asymptotic zero distribution of a class of hypergeometric polynomials

TL;DR

This paper analyzes the asymptotic distribution of zeros for a specific class of hypergeometric polynomials, revealing they tend to a lemniscate-shaped curve as the degree increases.

Contribution

It provides the first detailed asymptotic zero distribution result for this hypergeometric polynomial class, linking it to boundary cases in Jacobi polynomial classifications.

Findings

Zeros asymptotically approach a lemniscate section

Results connect to boundary cases in Jacobi polynomial asymptotics

Uses asymptotic analysis to describe zero distribution

Abstract

We prove that the zeros of ${}_2F_1(-n,\frac{n+1}{2};\frac{n+3}{2};z)$ asymptotically approach the section of the lemniscate $\{z: |z(1-z)^2|=4/27; \textrm{Re}(z)>1/3\}$ as $n\rightarrow \infty$. In recent papers (cf. \cite{KMF}, \cite{orive}), Mart\'inez-Finkelshtein and Kuijlaars and their co-authors have used Riemann-Hilbert methods to derive the asymptotic zero distribution of Jacobi polynomials $P_n^{(\alpha_n,\beta_n)}$ when the limits $\ds A=\lim_{n\rightarrow \infty}\frac{\alpha_n}{n}$ and $\ds B=\lim_{n\rightarrow \infty}\frac{\beta_n}{n}$ exist and lie in the interior of certain specified regions in the $AB$-plane. Our result corresponds to one of the transitional or boundary cases for Jacobi polynomials in the Kuijlaars Mart\'inez-Finkelshtein classification.