Asymptotic distributions of the zeros of certain classes of gauss hypergeometric polynomials

Abstract

Abstract We study the asymptotic distribution of the zeros of certain classes of Gauss hypergeometric polynomials. Classical analytic methods such as Watson’s lemma and the method of steepest descents are used to analyze the large-n behavior of the zeros of the Gauss hypergeometric polynomials: 2 F 1 ( - n , a ; kn + b ; z ) , 2 F 1 ( - n , kn + c ; kn + c + τ ; z ) and 2 F 1 ( - n , λ ; - pn + ν ; z ) , where n is a nonnegative integer, the constants k, a, τ, λ > 0, p > 1, and b , c , ν ∈ R . Owing to the connection between Jacobi polynomials and Gauss hypergeometric polynomials, we prove a special case of a conjecture made by Martinez–Finkelshtein, Martinez–Gonzalez, and Orive. Numerical evidence and graphical illustrations of the clustering of zeros on certain curves are generated by Mathematica (Version 4.0).