Eigenvalue Integro-Differential Equations for Orthogonal Polynomials on the Real Line∗
نویسندگان
چکیده
The one-dimensional harmonic oscillator wave functions are solutions to a SturmLiouville 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 (−∞,∞) 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. ∗PACS numbers: 02.60.Nm, 05.30.Fk, 21.60.Jz
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