Asymptotic Properties of Zeros of Hypergeometric Polynomials

Peter L. Duren; Bertrand J. Guillou

doi:10.1006/jath.2001.3580

Asymptotic Properties of Zeros of Hypergeometric Polynomials

Peter L. Duren, Bertrand J. Guillou

Mathematics

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

In a paper by K. Driver and P. Duren (1999, Numer. Algorithms21, 147-156) a theorem of Borwein and Chen was used to show that for each k∈N the zeros of the hypergeometric polynomials F(-n, kn+1; kn+2; z) cluster on the loop of the lemniscate {z:z^k(1-z)=k^k/(k+1)^k+1}, with Re{z}>k/(k+1) as n→∞. We now supply a direct proof which generalizes this result to arbitrary k>0, while showing that every point of the curve is a cluster point of zeros. Examples generated by computer graphics suggest some finer asymptotic properties of the zeros.

Original language	English
Pages (from-to)	329-343
Number of pages	15
Journal	Journal of Approximation Theory
Volume	111
Issue number	2
DOIs	https://doi.org/10.1006/jath.2001.3580
State	Published - Aug 2001

Bibliographical note

Funding Information:
1Both authors acknowledge support from the National Science Foundation. The paper is an outgrowth of a Research Experience for Undergraduates project at the University of Michigan.

Keywords

Asymptotics
Hypergeometric polynomials
Zeros
lemniscates

ASJC Scopus subject areas

Analysis
Numerical Analysis
General Mathematics
Applied Mathematics

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10.1006/jath.2001.3580

Cite this

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title = "Asymptotic Properties of Zeros of Hypergeometric Polynomials",

abstract = "In a paper by K. Driver and P. Duren (1999, Numer. Algorithms21, 147-156) a theorem of Borwein and Chen was used to show that for each k∈N the zeros of the hypergeometric polynomials F(-n, kn+1; kn+2; z) cluster on the loop of the lemniscate {z:zk(1-z)=kk/(k+1)k+1}, with Re{z}>k/(k+1) as n→∞. We now supply a direct proof which generalizes this result to arbitrary k>0, while showing that every point of the curve is a cluster point of zeros. Examples generated by computer graphics suggest some finer asymptotic properties of the zeros.",

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AU - Guillou, Bertrand J.

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AB - In a paper by K. Driver and P. Duren (1999, Numer. Algorithms21, 147-156) a theorem of Borwein and Chen was used to show that for each k∈N the zeros of the hypergeometric polynomials F(-n, kn+1; kn+2; z) cluster on the loop of the lemniscate {z:zk(1-z)=kk/(k+1)k+1}, with Re{z}>k/(k+1) as n→∞. We now supply a direct proof which generalizes this result to arbitrary k>0, while showing that every point of the curve is a cluster point of zeros. Examples generated by computer graphics suggest some finer asymptotic properties of the zeros.

KW - Asymptotics

KW - Hypergeometric polynomials

KW - Zeros

KW - lemniscates

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Asymptotic Properties of Zeros of Hypergeometric Polynomials

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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