On polynomial solutions of certain finite order ordinary differential equations

TL;DR

This paper investigates polynomial solutions of finite order differential equations, deriving explicit eigenfunctions and eigenvalues, and explores transformations and conditions for eigenfunction sequences, with applications to Hermite polynomials.

Contribution

It provides explicit forms of polynomial eigenfunctions and eigenvalues for certain differential operators and analyzes transformations affecting these eigenfunctions.

Findings

Explicit polynomial eigenfunctions and eigenvalues derived

Necessary conditions for transformed sequences to remain eigenfunctions

Application to classical Hermite polynomials

Abstract

Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial eigenfunctions are explicitly obtained, as well as the corresponding eigenvalues. Also, some linear transformations are applied to sequences of eigenfunctions and a necessary condition for this to be a sequence of eigenfunctions of a new differential operator is obtained. These results are applied to the particular case of classical Hermite polynomials.