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Zeros of a certain class of Gauss hypergeometric polynomials. (English) Zbl 1499.33018

Summary: We prove that as \(n\to\infty\), the zeros of the polynomial \[ _2F_1\left [\begin{matrix} -n,\alpha n+1\\ \alpha n+2\end{matrix} ;z\right] \] cluster on (a part of) a level curve of an explicit harmonic function. This generalizes previous results of K. Boggs and P. Duren [Comput. Methods Funct. Theory 1, No. 1, 275–287 (2001; Zbl 1009.33004)], K. Driver and P. Duren [Numer. Algorithms 21, No. 1–4, 147–156 (1999; Zbl 0935.33004)], and P. L. Duren and B. J. Guillou [J. Approx. Theory 111, No. 2, 329–343 (2001; Zbl 0983.33008)] to the case of a complex parameter \(\alpha\) and partially proves a conjecture made by the authors in an earlier work [Comput. Methods Funct. Theory 16, No. 2, 167–185 (2016; Zbl 1339.33009)].

MSC:

33C05 Classical hypergeometric functions, \({}_2F_1\)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
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