New transformation law for Heun and Hypergeometric Equations

TL;DR

This paper introduces novel non-linear transformation methods for Heun and Hypergeometric equations, expanding the toolkit for solving and relating these complex differential equations.

Contribution

It presents new forms of transformations between Heun and Hypergeometric equations, including non-Fuchsian cases, broadening the scope of equation solvability.

Findings

New Heun-to-Heun transformation forms

Heun-to-Hypergeometric transformations via non-linear variable change

Examples of transformations between non-Fuchsian equations

Abstract

In this work we establish new forms of Heun-to-Heun transformations and Heun-to-Hypergeometric transformations. The transformations are realised by changing the independent variable in a non-linear way. Using these we also point out some simple examples of transformations between equations that are not Fuchsian and that generalise the Heun-to-Hypergeometric transformations.