Polynomial solutions of differential equations

Abstract

A new approach for investigating polynomial solutions of differential equations is proposed. It is based on elementary linear algebra. Any differential operator of the form , where a k is a polynomial of degree ≤ k, over an infinite field F has all eigenvalues in F in the space of polynomials of degree at most n, for all 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. Specializing to the real field, the potential of the method is illustrated by recovering Bochner's classification of second order ODEs with polynomial coefficients and polynomial solutions, as well as cases missed by him - namely that of Romanovski polynomials, which are of recent interest in theoretical physics, and some Jacobi type polynomials. An important feature of this approach is the simplicity with which the eigenfunctions and their orthogonality and norms can be determined, resulting in significant reduction in computational complexity of such problems. 2000 MSC: 33C45; 34A05; 34A30; 34B24.