Eigenvalue Integro-Differential Equations for Orthogonal Polynomials on the Real Line

TL;DR

This paper introduces a new integro-differential equation framework to characterize orthogonal polynomials on the real line, extending beyond classical Sturm-Liouville problems, with specific application to Hahn-Meixner polynomials.

Contribution

It generalizes the classical Sturm-Liouville approach by formulating integro-differential equations for arbitrary orthogonal polynomials on the real line.

Findings

Characterization of orthogonal polynomials via integro-differential equations

Extension of Sturm-Liouville theory to a broader class of polynomials

Application to Hahn-Meixner polynomials

Abstract

The one-dimensional harmonic oscillator wave functions are solutions to a Sturm-Liouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained from a Sturm-Liouville problem on the whole real line. In this paper we show how to characterize an arbitrary set of polynomials orthogonal on $(-\infty,\infty)$ in terms of a system of integro-differential equations of Hartree-Fock type. This system replaces and generalizes the linear differential equation associated with a Sturm-Liouville problem. We demonstrate our results for the special case of Hahn-Meixner polynomials.