Novel representation of the general Heun's functions. Back to the beginning

TL;DR

This paper introduces a new symmetric series solution for the general Heun's equation, expanding the symmetry group and enabling a unified approach to all singular points.

Contribution

It presents a novel symmetric form of solutions for the general Heun's equation, extending its symmetry group beyond the Mobius group.

Findings

Derived the symmetry group of the symmetric form of Heun's equation.

Developed a series solution treating all singular points equally.

Extended the symmetry group beyond traditional Mobius transformations.

Abstract

We study a novel type of solutions of the general Heun's equation, based on its symmetric form. We derive the symmetry group of this equation which is a proper extension of the Mobius group. The new series solution treat simultaneously and on an equal footing all singular points.