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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic distributions of the zeros of a family of hypergeometric polynomials


Authors: Jian-Rong Zhou, H. M. Srivastava and Zhi-Gang Wang
Journal: Proc. Amer. Math. Soc. 140 (2012), 2333-2346
MSC (2010): Primary 33C05, 33C20; Secondary 30C15, 33C45
Published electronically: November 17, 2011
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Abstract: The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Gauss hypergeometric polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Gauss hypergeometric polynomials,

$\displaystyle \;_2F_1(-n, a; -n+b;z),$

where $ n$ is a nonnegative integer. Owing to the connection between the classical Jacobi polynomials and the Gauss hypergeometric polynomials, we prove a special case of a conjecture made by Martínez-Finkelshtein, Martínez-González and Orive. Numerical evidence and graphical illustrations of the clustering of the zeros on certain curves are generated by Mathematica (Version 4.0).

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Additional Information

Jian-Rong Zhou
Affiliation: Department of Mathematics, Foshan University, Foshan 528000, Guangdong Province, People’s Republic of China
Email: zhoujianrong2012@163.com

H. M. Srivastava
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
Email: harimsri@math.uvic.ca

Zhi-Gang Wang
Affiliation: School of Mathematics and Computing Science, Changsha University of Science and Technology (Yuntang Campus), Changsha 410114, Hunan Province, People’s Republic of China
Email: wangmath@163.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11117-1
PII: S 0002-9939(2011)11117-1
Keywords: Gauss hypergeometric function and polynomials, asymptotic distribution of zeros, zeros of $_{2}F_{1}(-n, a; -n+b;z)$, Jacobi polynomials, hypergeometric reduction formulas, Euler-Mascheroni constant, Vitali’s theorem, Hurwitz’s theorem, Eneström-Kakeya theorem, hypergeometric identity, Mathematica (Version 4.0).
Received by editor(s): March 26, 2010
Received by editor(s) in revised form: August 10, 2010, and February 14, 2011
Published electronically: November 17, 2011
Communicated by: Walter Van Assche
Article copyright: © Copyright 2011 American Mathematical Society