Generalized Bessel functions: Theoretical relevance, and computational techniques (Q2781660)

This article surveys some of the author's research on orthogonal polynomials with respect to a singular continuous measure, and their Fourier transforms with respect to this measure. For the (absolutely continuous) measure \((1-x^2)^{-1/2}dx\), these would be the Chebyshev polynomials, and their Fourier transforms would be Bessel functions, hence the title. Orthogonal polynomials arise in the spectral theory of a semi-infinite Hermitian tridiagonal matrix \(H\), and their Fourier transforms arise in solving the initial value problem for the Schrödinger equation \(i d\psi/dt=H\psi\). The author discusses a number of numerical algorithms for efficient calculation of the generalized Bessel functions obtained in this framework, and surveys relevant theoretical results and conjectures about their behavior.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00015].