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arXiv · hep-th/9411040

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

Carl M. Bender·Joshua Feinberg
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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.

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Carl M. Bender, Joshua Feinberg. 1994-11-04. Eigenvalue Integro-Differential Equations for Orthogonal Polynomials on the Real Line. https://doi.org/10.1063/1.531016

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