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

Srivastava, H.; Zhou, Jian-Rong; Wang, Zhi-Gang

doi:10.1090/S0025-5718-2011-02409-9

Asymptotic distributions of the zeros of certain classes of hypergeometric functions and polynomials
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by H. M. Srivastava, Jian-Rong Zhou and Zhi-Gang Wang PDF

Math. Comp. 80 (2011), 1769-1784 Request permission

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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