Polynomial Solutions of Differential Equations

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 $\le k$, 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.

Finite orthogonality of Romanovski polynomials