Asymptotic zero distribution of hypergeometric polynomials. (English) Zbl 0935.33004

Driver, Kathy; Duren, Peter

Numer. Algorithms 21, No. 1-4, 147-156 (1999).

The paper is devoted to the asymptotic zero distribution of hypergeometric polynomials of the form \[ F(-n,kn+1; (k+l)n+2;z), \qquad k,l,n\in \mathbb{N}. \] The equations of the curves are given, on which the zeros lie asymptotically as \(n\to\infty\). Furthermore it is shown that for \(l=0\) the zeros cluster on the loop of a suitable lemniscate \(L_k\) as \(n\to\infty\). Similar results are presented for other functions related to hypergeometric polynomials, including Jacobi polynomials and associated Legendre functions.

Reviewer: J.Saurer (Regensburg)

MSC:

33C20	Generalized hypergeometric series, \({}_pF_q\)
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

asymptotic zero distribution; hypergeometric polynomials; Jacobi polynomials; Legendre functions