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Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. (English) Zbl 0830.33001
The aim of this interesting paper is clearly and exhaustively described by the following authors’ abstract: “The nodal structure of the eigenfunctions of a large class of quantum mechanics potentials is fully described by the properties of the real zeros of the Gauss and Kummer hypergeometric functions, denoted by F(a,b;c;x) and M(a,b;c;x) respectively. Although numerous properties of an individual nature have been published in the literature, almost no property of global character is known. Here the distribution of zeros of these functions is explicitly found in terms of a-, b- and c-parameters. This is done by means of a recent semiclassical or WKB-like approach whose starting point is the second-order differential equation that these functions satisfy. The resulting approximate expression allows us to find the exact asymptotic distribution of zeros of the Jacobi and Laguerre polynomials, which are instances of the aforementioned hypergeometric functions”.
33C05Classical hypergeometric functions, 2 F 1
33C45Orthogonal polynomials and functions of hypergeometric type
33C15Confluent hypergeometric functions, Whittaker functions, 1 F 1