Asymptotic distributions of the zeros of certain classes of hypergeometric functions and polynomials

Abstract

Abstract. The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Clausenian hypergeometric 3 F 2 functions and polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Clausenian hypergeometric polynomials:where n is a nonnegative integer. Some families of the hypergeometric 3 F 2 functions, which are connected (by means of a hypergeometric reduction formula) with the Gauss hypergeometric polynomials of the form 2 F 1 (−n, kn + l + 1; kn + l + 2; z), are also investigated. Numerical evidence and graphical illustrations of the clustering of zeros on certain curves are generated by Mathematica (Version 4.0).

Quasi-orthogonality and real zeros of some F22 and FJohnston1, Jordaan2 2015Applied Numerical Mathematics3000View full textAdd to dashboardCite

“…[6], [8], [11] and [12]) have been obtained. The asymptotic behaviour of the zeros of certain families of 3 F 2 functions were studied by Driver and Jordaan in [9] using Hurwitz theorem with [24] by Srivastava, Zhou and Wang completing this work. In [10], Driver, Jordaan and Martínez-Finkelstein prove that the zeros of some families of 3 F 2 hypergeometric polynomials are all real and negative using the theory of Pólya frequency sequences and functions.…”

Zeros of hypergeometric functions in the p-adic setting

“…Indeed it has not been explored completely, and even when b, c are real. The zero location of special classes of such polynomials with restrictions on the parameters b and c can be found in [1, 3-5, 7, 8] and the asymptotic zero distribution of certain classes have been investigated in [2,6,9,10,22]. In view of these results, it is natural to study zeros of other hypergeometric type functions and investigate their consequences.…”

Zeros of hypergeometric functions in the $p$-adic setting

“…Indeed it has not been explored completely, and even when b, c are real. The zero location of special classes of such polynomials with restrictions on the parameters b and c can be found in [1,3,4,5,7,8] and the asymptotic zero distribution of certain classes have been investigated in [2,6,9,10,22]. These give motivation to study zeros of p-adic analogue of hypergeometric functions from number theoretic point of view.…”