Vector Orthogonal Polynomials with Bochner’s Property

Abstract

Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre, and Bessel have the property that the polynomials of each system are eigenfunctions of a second-order ordinary differential operator. According to a classical theorem by Bochner, they are the only systems with this property. Similarly, the polynomials of Charlier, Meixner, Kravchuk, and Hahn are both orthogonal and are eigenfunctions of a suitable difference operator of second order. We recall that according to the famous theorem of Favard–Shohat, the condition of orthogonality is equivalent to the 3-term recurrence relation. Vector orthogonal polynomials (VOP) satisfy finite-term recurrence relation with more terms, according to a theorem by J. Van Iseghem, and this characterizes them. Motivated by Bochner’s theorem, we are looking for VOP that are also eigenfunctions of a differential (difference) operator. We call these simultaneous conditions Bochner’s property. The goal of this paper is to introduce methods for construction of VOP which have Bochner’s property. The methods are purely algebraic and are based on automorphisms of noncommutative algebras. They also use ideas from the so-called bispectral problem. Applications of the abstract methods include broad generalizations of the classical orthogonal polynomials, both continuous and discrete. Other results connect different families of VOP, including the classical ones, by linear transforms of purely algebraic origin, despite of the fact that, when interpreted analytically, they are integral transformations.