  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.

High Energy Physics - Theory;Condensed Matter;Functional Analysis;Functional Analysis