Heun functions versus elliptic functions

TL;DR

This paper reviews recent advances in Heun functions, connecting classical analysis, elliptic functions, and their representations, including Picard's generalization of Floquet theory and elliptic integral forms.

Contribution

It introduces new representations of Heun functions using elliptic functions and extends Floquet theory to doubly periodic coefficients.

Findings

Heun functions can be expressed in terms of Jacobi theta functions.

Finite-gap solutions provide alternative integral representations.

Level one Heun functions are shown to be equivalent to elliptic forms.

Abstract

We present some recent progresses on Heun functions, gathering results from classical analysis up to elliptic functions. We describe Picard's generalization of Floquet's theory for differential equations with doubly periodic coefficients and give the detailed forms of the level one Heun functions in terms of Jacobi theta functions. The finite-gap solutions give an interesting alternative integral representation which, at level one, is shown to be equivalent to their elliptic form.