On polynomial solutions of differential equations

TL;DR

This paper introduces a general method for deriving linear differential equations with polynomial solutions, linking algebraic structures like Lie algebras and superalgebras to polynomial solutions of differential and difference equations.

Contribution

It presents a novel approach connecting spectral problems in Lie algebra representations to polynomial solutions of differential equations, including classical and quantum cases.

Findings

Polynomial solutions occur in partial differential equations and matrix differential equations.

Classical orthogonal polynomials emerge from $SL(2,\mathbb{R})$ actions.

Examples include polynomials related to $sl_2(\mathbb{R})$, $sl_2(\mathbb{R})_q$, $osp(2,2)$, and $so_3$.

Abstract

A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the "projectivized" representation possessing an invariant subspace and the spectral problem for a certain linear differential operator with variable coefficients. It is shown in general that polynomial solutions of partial differential equations occur; in the case of Lie superalgebras there are polynomial solutions of some matrix differential equations, quantum algebras give rise to polynomial solutions of finite--difference equations. Particularly, known classical orthogonal polynomials will appear when considering $SL(2,{\bf R})$ acting on ${\bf RP_1}$. As examples, some polynomials connected to projectivized representations of $sl_2 ({\bf R})$, $sl_2 ({\bf R})_q$, $osp(2,2)$ and $so_3$ are briefly discussed.