Spectral theory and special functions

TL;DR

This paper explores the application of spectral theory to special functions, demonstrating how spectral theorems for Jacobi operators yield orthogonality relations and insights into the moment problem.

Contribution

It introduces new applications of the spectral theorem to Jacobi operators on different spaces, linking spectral properties to orthogonal polynomials and hypergeometric functions.

Findings

Proof of Favard's theorem via spectral theorem

Explicit spectral measures for Meixner and Meixner-Pollaczek functions

Derived orthogonality relations for hypergeometric series

Abstract

A short introduction to the use of the spectral theorem for self-adjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on l^2(N), leading to a proof of Favard's theorem stating that polynomials satisfying a three-term recurrence relation are orthogonal polynomials. We discuss the link to the moment problem. In the second example, the spectral theorem is applied to Jacobi operators on l^2(Z). We discuss the theorem of Masson and Repka linking the deficiency indices of a Jacobi operator on l^2(Z) to those of two Jacobi operators on l^2(N). For two examples of Jacobi operators on l^2(Z), namely for the Meixner, respectively Meixner-Pollaczek, functions, related to the associated Meixner, respectively Meixner-Pollaczek, polynomials, and for the second order hypergeometric q-difference operator, we calculate the spectral measure explicitly. This gives explicit (generalised) orthogonality relations for hypergeometric and basic hypergeometric series.