An Extension of Bochner's Problem: Exceptional Invariant Subspaces

TL;DR

This paper extends Bochner's classical result by classifying exceptional polynomial subspaces, leading to new orthogonal polynomial systems as solutions of second-order eigenvalue equations with rational coefficients.

Contribution

It introduces a classification of exceptional invariant subspaces, enabling the construction of new orthogonal polynomial systems beyond classical families.

Findings

New families of orthogonal polynomials identified

Classification of exceptional polynomial subspaces achieved

Solutions involve second-order eigenvalue equations with rational coefficients

Abstract

A classical result due to Bochner characterizes the classical orthogonal polynomial systems as solutions of a second-order eigenvalue equation. We extend Bochner's result by dropping the assumption that the first element of the orthogonal polynomial sequence be a constant. This approach gives rise to new families of complete orthogonal polynomial systems that arise as solutions of second-order eigenvalue equations with rational coefficients. The results are based on a classification of exceptional polynomial subspaces of codimension one under projective transformations.