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Some recent results on the zeros of Bessel functions and orthogonal polynomials. (English) Zbl 0989.33004
This is a survey of the results on the zeros of Bessel functions and the classical orthogonal polynomials, obtained in the past 20 years specially by the author and specially derived from his long cooperation with A. Laforgia, M. E. Muldoon and P. Siafarikas. Section 1 of the survey is devoted to the zeros of the Bessel functions, and more generally of the cylinder functions. Standard methods of investigation of the properties of the zeros are recalled. Section 2 is concerned with the zeros of some orthogonal polynomials as Jacobi, ultraspherical and Hermite. This Section provides also information on the values of the Laguerre and Legendre polynomials. The selection of the results concentrates on those which are recent, or which have connections with results of other authors. The paper contains some relevant and interesting new results which appear here for the first time. Also open problems and conjectures are formulated.

MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1
33C45Orthogonal polynomials and functions of hypergeometric type