Vector orthogonal polynomials with Bochner's property

TL;DR

This paper develops algebraic methods to construct vector orthogonal polynomials that are eigenfunctions of differential or difference operators, extending classical orthogonal polynomial systems with Bochner's property.

Contribution

It introduces algebraic techniques based on automorphisms and bispectral ideas to generate VOP with Bochner's property, broadening classical polynomial families.

Findings

Constructed new classes of VOP with Bochner's property.

Established algebraic methods for generating VOP.

Connected classical and generalized VOP via algebraic transforms.

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 non-commutative 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.