Novel representation of the general Heun's functions

TL;DR

This paper introduces a new symmetric form of the general Heun's equation, expanding the solution space and symmetry group, aiming to simplify problem-solving and computational approaches in the theory of Heun's functions.

Contribution

It develops a novel symmetric form of the general Heun's equation and explores its symmetry group, providing new series solutions that treat all singular points equally.

Findings

Derived the symmetry group extending the Möbius group.

Introduced new series solutions for the symmetric form.

Potential to simplify solving and computing Heun's functions.

Abstract

In the present article we introduce and study a novel type of solutions of the general Heun's equation. Our approach is based on the symmetric form of the Heun's differential equation yielded by development of the Felix Klein symmetric form of the Fuchsian equations with an arbitrary number $N\geq 4$ of regular singular points. We derive the symmetry group of these equations which turns to be a proper extension of the Mobius group. We also introduce and study new series solution of symmetric form of the general Heun's differential equation (N=4) which treats simultaneously and on an equal footing all singular points. Hopefully, this new form will simplify the resolution of the existing open problems in the theory of general Heun's functions and can be used for development of new effective computational methods.