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

22 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.

Funding

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.

Funders
National Science Foundation Arctic Social Science Program

Keywords

Asymptotics
Hypergeometric polynomials
Zeros
lemniscates

ASJC Scopus subject areas

Analysis
Numerical Analysis
General Mathematics
Applied Mathematics

Access to Document

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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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.",

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T1 - Asymptotic Properties of Zeros of Hypergeometric Polynomials

AU - Duren, Peter L.

AU - Guillou, Bertrand J.

N1 - 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.

PY - 2001/8

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N2 - 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.

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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DO - 10.1006/jath.2001.3580

M3 - Article

AN - SCOPUS:0346609661

SN - 0021-9045

VL - 111

SP - 329

EP - 343

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 2

ER -

Asymptotic Properties of Zeros of Hypergeometric Polynomials

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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