On polynomial solutions of Heun equation

TL;DR

This paper introduces a new method to find polynomial solutions of the Heun equation and its extension, expanding the understanding of solutions to this complex differential equation.

Contribution

The authors develop a novel approach to identify polynomial solutions of the Heun equation and its extended form, broadening the solution space for these equations.

Findings

Identified a wide class of polynomial solutions to the Heun equation.

Constructed series solutions before deriving polynomial solutions.

Extended the method to a modified Heun equation with an additional term.

Abstract

By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before identifying the polynomial solutions. The Heun equation extended by the addition of a term, $- \s/x$, is also amenable for polynomial solutions.