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

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

1994-11-04

arxiv.org/abs/hep-th/9411040v1

J.Math.Phys. 36 (1995) 3106-3125

Physics

High Energy Physics

High Energy Physics - Theory

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

Scientific paper

10.1063/1.531016

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.

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