Examples of Heun and Mathieu functions as solutions of wave equations in curved spaces

TL;DR

This paper explores solutions to wave equations in curved spacetime backgrounds, showing how Mathieu functions and Heun functions naturally arise in different dimensional settings and providing transformations between them.

Contribution

It demonstrates the appearance of Heun functions in five-dimensional wave equations and connects them to Mathieu functions through explicit transformations.

Findings

Mathieu functions solve wave equations in four-dimensional curved spaces

Heun functions appear as solutions in five-dimensional cases

Transformations relate Heun functions to Mathieu functions

Abstract

We give examples where the Heun function exists in general relativity. It turns out that while a wave equation written in the background of certain metric yields Mathieu functions as its solutions in four space-time dimensions, the trivial generalization to five dimensions results in the double confluent Heun function. We reduce this solution to the Mathieu function with some transformations.