Spectral properties of solutions of hypergeometric-type differential equations (Q1334803)

Important problems in mathematical physics involving the Laplace and Helmholtz equations are connected to the differential equation \(\sigma(x) D^ 2 y+ \tau(x) Dy+ \lambda y=0\) whose solutions are known as hypergeometric-type functions (HTF's) when the coefficients \(\sigma(x)\) and \(\tau(x)\) are polynomials of respective degres at most two and one. The constant \(\lambda= -\nu \tau'- (1/2) \nu(\nu -1) \sigma''\); and, \(\sigma\) and \(\tau\) are independent of \(\nu\in \mathbb{R}\). The authors derive general analytic closed formulae that express the WKB density of the zeros of HTF's in terms of the coefficients. They study the spectral properties of these canonical forms and present interesting applications of their results in terms of the Gaussian and confluent hypergeometric functions, Legendre polynomials and Hermite functions.