Asymptotic zero distribution of a class of hypergeometric polynomials

Abstract

We prove that the zeros of ₂ F ₁ (−n, n+1/2; n+3/2; z asymptotically approach the section of the lemniscate {z : |z(1 − z)²&verbar = 4/27; Re(z) > 1/3} as n → ∞. In recent papers (cf. [8], [10]), Martínez-Finkelshtein and Kuijlaars and their co-authors have used Riemann-Hilbert methods to derive the asymptotic zero distribution of Jacobi polynomials P _n ^{(α_n , β_n )} when the limits A = lim _{n → ∞} α_n /n and B = lim _{n → ∞} β_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 Martínez-Finkelshtein classification.