d-Orthogonal Analogs of Classical Orthogonal Polynomials

Abstract

Classical orthogonal polynomial systems of Jacobi, Hermite, and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner, they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper, we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to d measures, rather than one. These polynomial systems enjoy a number of properties that make them a natural analog of the classical orthogonal polynomials. In the present paper, we continue their study. The most important new properties are their hypergeometric representations, which allow us to derive their generating functions and, in some cases, also Mehler-Heine type formulas.