The bispectral problem and polynomial solutions

TL;DR

This paper investigates the properties of polynomial solutions to the bispectral problem, explicitly constructing eigenfunctions and eigenvalues for differential operators, and exploring conditions for the inverse problem.

Contribution

It provides explicit constructions of polynomial eigenfunctions and eigenvalues, and establishes conditions for the existence of differential operators from given eigenpolynomials and eigenvalues.

Findings

Explicit polynomial eigenfunctions and eigenvalues derived

Conditions for the existence of differential operators from eigenpolynomials established

Properties and relations of polynomial solutions analyzed

Abstract

Some properties and relations satisfied by the polynomial solutions of the bispectral problem are studied. Given a differential operator, under certain restrictions its polynomial eigenfunctions are explicitly obtained, as well as the corresponding eigenvalues. Also the inverse problem is approached, giving conditions for the existence of a differential operator from its eigenpolynomials and eigenvalues.