Asymptotic distributions of the zeros of certain classes of hypergeometric functions and polynomials

Authors:
H. M. Srivastava, Jian-Rong Zhou and Zhi-Gang Wang

Journal:
Math. Comp. **80** (2011), 1769-1784

MSC (2010):
Primary 33C05, 33C20; Secondary 30C15, 33C45

DOI:
https://doi.org/10.1090/S0025-5718-2011-02409-9

Published electronically:
February 11, 2011

MathSciNet review:
2785478

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Clausenian hypergeometric $\;_3F_2$ functions and polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Clausenian hypergeometric polynomials: \[ \;_3F_2(-n, \tau n+a, b;\tau n+c, -n+d;z),\] where $n$ is a nonnegative integer. Some families of the hypergeometric $_3F_2$ functions, which are connected (by means of a hypergeometric reduction formula) with the Gauss hypergeometric polynomials of the form \[ \;_2F_1(-n,kn+l+1;kn+l+2;z),\] are also investigated. Numerical evidence and graphical illustrations of the clustering of zeros on certain curves are generated by *Mathematica* (Version 4.0).

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Additional Information

**H. M. Srivastava**

Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

Email:
harimsri@math.uvic.ca

**Jian-Rong Zhou**

Affiliation:
Department of Mathematics, Foshan University, Foshan 528000, Guangdong, People’s Republic of China

Email:
zhoujianrong2008@yahoo.com.cn

**Zhi-Gang Wang**

Affiliation:
School of Mathematics and Computing Science, Changsha University of Science and Technology, Yuntang Campus, Changsha 410114, Hunan, People’s Republic of China

Email:
wangmath@163.com

Keywords:
Generalized hypergeometric functions,
Clausenian hypergeometric function,
Gauss hypergeometric polynomials,
asymptotic distribution of zeros,
zeros of $_3F_2(-n, \tau n+a, b;\tau n+c, -n+d;z)$,
zeros of $_3F_2(-n,a,b;c,d;z)$,
Jacobi polynomials,
Rice polynomials,
Pasternack polynomials,
hypergeometric reduction formulas,
Euler-Mascheroni constant,
Vitali’s theorem,
Hurwitz’s theorem,
Eneström-Kakeya theorem,
*Mathematica* (Version 4.0).

Received by editor(s):
December 1, 2009

Received by editor(s) in revised form:
January 7, 2010

Published electronically:
February 11, 2011

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.