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Mathematics of Computation

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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
Published electronically: February 11, 2011
MathSciNet review: 2785478
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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 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

Jian-Rong Zhou
Affiliation: Department of Mathematics, Foshan University, Foshan 528000, Guangdong, People鈥檚 Republic of China

Zhi-Gang Wang
Affiliation: School of Mathematics and Computing Science, Changsha University of Science and Technology, Yuntang Campus, Changsha 410114, Hunan, People鈥檚 Republic of China

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鈥檚 theorem, Hurwitz鈥檚 theorem, Enestr枚m-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.