An extension of Bochner's problem: Exceptional invariant subspaces

A classical result due to Bochner characterizes the classical orthogonal polynomial systems as solutions of a secondorder 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.