Eigenvalue integrodifferential equations for orthogonal polynomials on the real line

Carl M. Bender, Joshua Feinberg

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)3106-3125
Number of pages20
JournalJournal of Mathematical Physics
Issue number6
StatePublished - 1995
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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