Eigenvalue integrodifferential equations for orthogonal polynomials on the real line

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 it is shown how to characterize an arbitrary set of polynomials orthogonal on ( -∞,∞) in terms of a system of integrodifferential equations of the Hartree-Fock type. This system replaces and generalizes the linear differential equation associated with a Sturm-Liouville problem. Our results are demonstrated for the special case of continuous Hahn polynomials.