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Asymptotic distribution of zeros of a certain class of hypergeometric polynomials. (English) Zbl 1339.33009
Summary: We study the asymptotic behavior of the zeros of a family of a certain class of hypergeometric polynomials \(_A\mathrm {F}_B\left[\begin{matrix}{-n,a_2,\dots,a_A}\\{b_1,b_2,\dots,b_B}\end{matrix} ; z \right] \), using the associated hypergeometric differential equation, as the parameters go to infinity. The curve configuration on which the zeros cluster is characterized as level curves associated with integrals on an algebraic curve. The algebraic curve is the hypergeometrc differential equation, using a similar approach to the method used in J. Borcea et al. [Publ. Res. Inst. Math. Sci. 45, No. 2, 525–568 (2009; Zbl 1182.30008); corrigendum 48, No. 1, 229–233 (2012)]. In a specific degenerate case, we make a conjecture that generalizes work in 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)], and present experimental evidence to substantiate it.

MSC:
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C20 Generalized hypergeometric series, \({}_pF_q\)
31A35 Connections of harmonic functions with differential equations in two dimensions
34E05 Asymptotic expansions of solutions to ordinary differential equations
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