On the classification of hypergeometric families of orthogonal polynomials on the real line
Joseph Bernstein, Dmitry Gourevitch, Siddhartha Sahi
TL;DR
This work introduces and classifies hypergeometric families of orthogonal polynomials on the real line (HG-families) without assuming a fixed second-order operator. Using an algebraic framework of 1-dimensional A-modules and Gauss–Favard theory, it shows that any orthogonal HG-family arises from a pair $(R,f)$ with $ ext{deg}\,R\le2$ and $f$ possessing a precise rational-hypergeometric form, yielding exactly 10 HG-types: 8 from the Askey scheme and 2 new Lommel-based families expressible as $_4F_1$. It extends the theory to quasi-orthogonal and rational HG-families, providing a full structure theorem, explicit parameterizations, and new examples including the families ${E_n^{(c)}}$ and ${F_n^{(c)}}$ tied to Lommel polynomials, along with discrete measures and fourth-order differential equations. The results generalize across any field of characteristic zero and connect classical hypergeometric families to novel rational-HG constructions, enriching the landscape of orthogonal polynomials and their hypergeometric representations. Overall, the paper delivers a comprehensive, algebraic classification beyond differential-eigenfunction constraints and introduces new families with potential mathematical and physical applications.
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.