On the classification of hypergeometric families of orthogonal polynomials on the real line

TL;DR

This paper classifies all hypergeometric families of orthogonal polynomials on the real line, revealing exactly 10 types, including well-known and newly identified families, using an algebraic approach that generalizes previous classifications.

Contribution

It introduces a new, more general definition of hypergeometric families and provides a complete algebraic classification of all such orthogonal polynomial families.

Findings

Exactly 10 types of hypergeometric orthogonal polynomial families identified

8 from the Askey scheme and 2 new Lommel polynomial-based families

Classification valid over any field of characteristic zero

Abstract

Several important families of orthogonal polynomials on the real line are called ``hypergeometric'' since they can be explicitly described in terms of some hypergeometric series $_pF_q$ that uses the degree $n$ of the polynomial as a parameter. It is natural to ask if one can classify all such families. Indeed many classification results have been obtained in this direction, but only under the additional assumption that the polynomials are eigenfunctions of some second order operator. In this paper we initiate a new approach to this classification. We propose a definition of an HG family that makes precise, but also generalizes, the notion of a ``hypergeometric'' family. Our main result is that there are exactly 10 types of orthogonal HG families, 8 from the well-known Askey scheme and 2 additional types of families that can be expressed in terms of Lommel polynomials. Our methods in this paper are algebraic. In particular, we classify a wider class of quasi-orthogonal HG families, and this classification is valid over an arbitrary field of characteristic zero. We also define a more general class of rational HG families and prove a structure theorem for quasi-orthogonal families in this class. We provide examples for such families, that are in particular new families of orthogonal polynomials of potential interest.