Asymptotic zero distribution of hypergeometric polynomials

Abstract

We show that the zeros of the hypergeometric polynomials , , cluster on the loop of the lemniscate {k mathord{left/{vphantom {k {left( {k + 1} right)}}} right.kern-nulldelimiterspace} {left( {k + 1} right)}}}}} right.kern-nulldelimiterspace} {left( {k + 1} right)^{k + 1} ,{text{Re}}left( z right) > {k mathord{left/{vphantom {k {left( {k + 1} right)}}} right.kern-nulldelimiterspace} {left( {k + 1} right)}}}}} right}$$]]> as infty$$]]> . We also state the equations of the curves on which the zeros of , lie asymptotically as infty$$]]> . Auxiliary results for the asymptotic zero distribution of other functions related to hypergeometric polynomials are proved, including Jacobi polynomials with varying parameters and associated Legendre functions. Graphical evidence is provided using Mathematica.