A characterization of classical orthogonal Laurent polynomials

Abstract

In [3] certain Laurent polynomials of ₂F₁ genus were called “Jacobi Laurent polynomials”. These Laurent polynomials belong to systems which are orthogonal with respect to a moment sequence ((a)_n/(c)_n)_nεℤ where a, c are certain real numbers. Together with their confluent forms, belonging to systems which are orthogonal with respect to 1/(c)_n)_nεℤ respectively ((a)_n)_nεℤ, these Laurent polynomials will be called “classical”. The main purpose of this paper is to determine all the simple (see section 1) orthogonal systems of Laurent polynomials of which the members satisfy certain second order differential equations with polynomial coefficients, analogously to the well known characterization of S. Bochner [1] for ordinary polynomials.