Asymptotic distributions of the zeros of a family of hypergeometric polynomials

Abstract

The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Gauss hypergeometric polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Gauss hypergeometric polynomials, \[ 2 F 1 ( − n , a ; − n + b ; z ) , \;_2F_1(-n, a; -n+b;z), \] where n n is a nonnegative integer. Owing to the connection between the classical Jacobi polynomials and the Gauss hypergeometric polynomials, we prove a special case of a conjecture made by Martínez-Finkelshtein, Martínez-González and Orive. Numerical evidence and graphical illustrations of the clustering of the zeros on certain curves are generated by Mathematica (Version 4.0).