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Asymptotic Distribution of Zeros of a Certain Class of Hypergeometric Polynomials

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Abstract

We study the asymptotic behavior of the zeros of a family of a certain class of hypergeometric polynomials \(\small {}_{A}\text {F}_{B}\left[ \begin{array}{c} -n,a_2,\ldots ,a_A \\ b_1,b_2,\ldots ,b_B \end{array} ; \begin{array}{cc} z \end{array}\right] \), 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 Borcea et al. (Publ Res Inst Math Sci 45(2):525–568, 2009). In a specific degenerate case, we make a conjecture that generalizes work in Boggs and Duren (Comput Methods Funct Theory 1(1):275–287, 2001), Driver and Duren (Algorithms 21(1–4):147–156, 1999), and Duren and Guillou (J Approx Theory 111(2):329–343, 2001), and present experimental evidence to substantiate it.

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Authors and Affiliations

  1. Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia

    Addisalem Abathun

  2. Department of Mathematics, Stockholm University, Stockholm, Sweden

    Addisalem Abathun & Rikard Bøgvad

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  1. Addisalem Abathun
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  2. Rikard Bøgvad
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Correspondence to Addisalem Abathun.

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Communicated by Stephan Ruscheweyh.

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Abathun, A., Bøgvad, R. Asymptotic Distribution of Zeros of a Certain Class of Hypergeometric Polynomials. Comput. Methods Funct. Theory 16, 167–185 (2016). https://doi.org/10.1007/s40315-015-0131-1

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  • DOI: https://doi.org/10.1007/s40315-015-0131-1

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