From Heun Class Equations to Painlev\'e Equations

TL;DR

This paper explores the connections between Heun class differential equations and Painlevé equations, providing a unified classification and introducing deformed Heun equations linked to all Painlevé types.

Contribution

It offers a unified framework for classifying Heun and Painlevé equations, including deformed versions with additional singularities, and discusses their interrelations.

Findings

Classification of Heun equations into 5 supertypes and 10 types.

Establishment of a direct relationship between deformed Heun equations and Painlevé equations.

Unified treatment of different 'time variables' in Painlevé equations.

Abstract

In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlev\'e I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonlogarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlev\'e equations. In particular, Painlev\'e equations can be also divided into 5 supertypes, and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the ''time variable'' may depend on the type. We describe unified treatments for several possible ''time variables''.