Polynomial Solutions of Differential Equations

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.

where k a is a real polynomial of degree 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 n x in ( ) n L x . 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 nonnegative 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.
The subject of polynomial solutions of differential equations is a classical theme, going back to Routh [10] and Bochner [3]. A comprehensive survey of recent literature is given in [6]. One family of polynomials-namely the Romanovski polynomials [4,9] is missing even in recent mathematics literature on the subject [8]; these polynomials are the main subject of some current Physics literature [9,11]. Their existence and -under a mild condition -uniqueness and orthogonality follow from the following propositions. The proofs use elementary linear algebra and are suitable for class-room exposition. The same ideas work for higher order equations [1].

Proposition 1
and L has eigenfunctions in each j P .
Assume that the eigenvalues of L are distinct. Then n P has a basis of eigenfunctions and, for reasons of degree, there must be an eigenfunction of degree n, for every n. Therefore, up to a constant, there is a unique eigenfunction of degree n for all n.
We now concentrate on second order operators, leaving the higher order case to [1]. Let then by scaling and translation, we may assume that 1 , 1 ) ( x . Applying the above proposition we then have the following result.

Proposition 2
, then the eigenspace in n P for eigenvalue n n n α λ In this proposition there is no claim to any kind of orthogonality properties. Nevertheless, the non-classical functions appearing here are of great interest in Physics and their properties and applications are investigated in [4,9,11].
The classical Legendre, Hermite, Laguerre and Jacobi make their appearance as soon as one searches for self-adjoint operators. Their existence and orthogonality properties [cf:8, p.80-106,2,7] can be obtained elegantly in the context of elementary Sturm-Liouville theory.

Proposition 3
Let L be the operator defined by ( ) functions which are at least two times differentiable on a finite interval I.

Define a bilinear function on C by
where p is two times differentiable and non-negative and does not vanish identically in any subinterval of I.

Proof:
Let , Equating coefficients of and u u′ on both sides, we get the differential equations for p: Examples: (1) Jacobi polynomials First note that for any differentiable function f with f ′ continuous, the integral α < -as one sees by using integration by parts.
Consider the equation , then L would be a self-adjoint operator on all polynomials of degree n and so, there must be, up to a scalar, a unique polynomial which is an eigen function of L for eigenvalue ( 1) n n nα − − + .
So these polynomials satisfy the equation (2) The equation This equation is investigated in [5] and the eigenvalues determined experimentally, by machine computations. Here, we will determine the eigenvalues in the framework provided by Proposition 3.
. L e t n P be the space of al polynomials of degree at most n. As L maps n P into itself, the eigenvalues of L are given by the . The eigenvalues turn out to be 2 n − . As these eigenvalues are distinct, there is, up to a constant, a unique polynomial of degree n which is an eigenfunction of L.

The weight function is
on the interval [0,1] and it is not integrable. However, as 0 , the operator maps the space V of all polynomials that are multiples of ) The requirement for L to be self-adjoint on V is 0 As η ξ , vanish at 1, the operator L is indeed self-adjoint on V.
, where n P is the space of al polynomials of degree at most n.
As the codimension of n V in 1 + n V is 1, the operator L must have an eigenvector in n V for all the degrees from 1 to ) 1 ( + n . Therefore, up to a scalar, there is a unique eigenfunction of degree ( 1 + n ) which is a multiple of ) 1 ( t − and all these functions are orthogonal for the weight Using the uniqueness up to scalars of these functions, the eigenfunctions are determined by the differential equation and can be computed explicitly.

(3) The Finite Orthogonality of Romanovski Polynomials
These polynomials are investigated in Refs [11,9] and their finite orthogonality is proved also proved there. Here, we establish this in the framework of

Proposition3.
The Romanovski polynomials are eigenfunctions of the operator not an integer, there is only one monic polynomial in every degree which is an eigenfunction of L ; for α a non-positive integer, the eigenspaces can be 2 dimensional for certain degrees (Propostion2).
The formal weight function is For several non-trivial applications to problems in Physics, the reader is referred to the paper [9].

Conclusion:
In this note, which should have been written at least hundred years ago, we have rederived several results from classical and recent literature from a unified point of view by a straightforward application of basic linear algebra.
Some of these polynomials are not discussed in the standard textbooks on the subject, e.g. [8]-as pointed out in Ref [9].
We have also derived the orthogonality-classical as well as finite-of these polynomials from a unified point of view.