Polynomial Solutions of Differential Equations

TL;DR

This paper proves that differential operators with polynomial coefficients have all real eigenvalues on polynomial spaces and characterizes their eigenfunctions, extending classical results and identifying new polynomial families.

Contribution

It establishes the reality of eigenvalues for a broad class of polynomial differential operators and characterizes their eigenfunctions, generalizing Bochner's classification.

Findings

Eigenvalues are given by coefficients in polynomial action

Unique monic eigenpolynomials exist for distinct eigenvalues

Reveals new families of non-classical polynomials

Abstract

We show that any differential operator of the form $L(y)=\sum_{k=0}^{k=N} a_{k}(x) y^{(k)}$, where $a_k$ is a real polynomial of degree $\leq k$, has all real eigenvalues in the space of polynomials of degree at most n, for all n. The eigenvalues are given by the coefficient of $x^n$ in $L(x^{n})$. If these eigenvalues are distinct, then there is a unique monic polynomial of degree n which is an eigenfunction of the operator L- for every non-negative integer n. As an application we recover Bochner's classification of second order ODEs with polynomial coefficients and polynomial solutions, as well as a family of non-classical polynomials.