Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations

TL;DR

This paper establishes a connection between the Heun equations and Abel equations, providing explicit solutions and parameter relations, enhancing understanding of their solutions and integrability.

Contribution

It introduces a novel link between Heun equations and Abel equations, offering closed-form solutions and insights into their parameter spaces.

Findings

Closed-form solutions for GHE, CHE, BHE in terms of hypergeometric functions

Parameter relations that yield Liouvillian solutions

Mechanism relating equations with different numbers of singularities

Abstract

In a recent paper, the canonical forms of a new multi-parameter class of Abel differential equations, so-called AIR, all of whose members can be mapped into Riccati equations, were shown to be related to the differential equations for the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and Biconfluent (BHE) equations is presented. This connection fixes the value of one of the Heun parameters, expresses another one in terms of those remaining, and provides closed form solutions in terms of pFq functions for the resulting GHE, CHE and BHE, respectively depending on four, three and two irreducible parameters. This connection also turns evident what is the relation between the Heun parameters such that the solutions admit Liouvillian form, and suggests a mechanism for relating linear equations with N and N-1 singularities through the canonical forms of a non-linear equation of one order less.