Computational Methods and Function Theory

, Volume 16, Issue 2, pp 167–185

# Asymptotic Distribution of Zeros of a Certain Class of Hypergeometric Polynomials

Article

## 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.

## Keywords

Hypergeometric polynomials Cauchy transform Asymptotic zero measures Hypergeometric differential equation

## Mathematics Subject Classification

33C05 33C20 31A35 34E05

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