Applications of spectral theory to special functions

TL;DR

This paper explores how spectral theory of operators helps understand special functions, especially orthogonal polynomials, by analyzing their eigenfunction properties and spectral characteristics.

Contribution

It reviews existing literature on spectral analysis of operators related to special functions, highlighting applications to orthogonal polynomials and their matrix-valued counterparts.

Findings

Spectral properties provide explicit information about special functions.

Eigenfunction relations are central to understanding special functions.

Applications include analysis of orthogonal polynomials and their matrix analogues.

Abstract

Lecture notes for one of the courses at the OPSFA Summerschool 6, July 11-15, 2016. All the results in these notes have appeared in the literature. Many special functions are eigenfunctions to explicit operators, such as difference and differential operators, which is in particular true for the special functions occurring in the Askey-scheme, its $q$-analogue and extensions. The study of the spectral properties of such operators leads to explicit information for the corresponding special functions. We discuss several instances of this application, involving orthogonal polynomials and their matrix-valued analogues.