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Asymptotic zero distribution of hypergeometric polynomials

Authors:	Kathy Driver Peter Duren

Institution:	(1) Department of Mathematics, University of the Witwatersrand, P.O. Wits, 2050 Johannesburg, South Africa;(2) Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

Abstract:	We show that the zeros of the hypergeometric polynomials $F\left( { - n,kn + 1;kn + 2;z} \right)$ , $k,n \in \mathbb{N}$ , cluster on the loop of the lemniscate $\left\{ {z:\|z^k \left( {1 - z} \right)\| = {{k^k } \mathord{\left/{\vphantom {{k^k } {\left( {k + 1} \right)^{k + 1} ,{\text{Re}}\left( z \right) > {k \mathord{\left/{\vphantom {k {\left( {k + 1} \right)}}} \right.\kern-\nulldelimiterspace} {\left( {k + 1} \right)}}}}} \right.\kern-\nulldelimiterspace} {\left( {k + 1} \right)^{k + 1} ,{\text{Re}}\left( z \right) > {k \mathord{\left/{\vphantom {k {\left( {k + 1} \right)}}} \right.\kern-\nulldelimiterspace} {\left( {k + 1} \right)}}}}} \right\}$ as $n > \infty$ . We also state the equations of the curves on which the zeros of $F\left( { - n,kn + 1;\left( {k + \ell } \right)n + 2;z} \right),k,\ell ,n \in \mathbb{N}$ , lie asymptotically as $n > \infty$ . 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. This revised version was published online in June 2006 with corrections to the Cover Date.

Keywords:	hypergeometric polynomials asymptotic zero distribution
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