$d$-Orthogonal Analogs of Classical Orthogonal Polynomials

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 which 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.