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

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.

28 pages, Latex, U. Texas at Austin/ Washington University preprint

References in corpus (1)

• Stabilised Matrix Models for Non-Perturbative Two Dimensional Quantum Gravity

Cited by in corpus (2)

• Quasi-Exactly Solvable Systems and Orthogonal Polynomials
• The Bender-Dunne basis operators as Hilbert space operators