Heun equation and Painlev\'e equation

TL;DR

This paper explores the connection between sixth Painlevé solutions and finite-gap Heun solutions through monodromy analysis, providing new elliptic forms and differential formulas related to elliptic modular functions.

Contribution

It establishes a novel link between Painlevé and Heun equations via monodromy, and derives elliptic forms of Painlevé equations with explicit differential formulas.

Findings

Relation between Painlevé and Heun solutions established

Elliptic form of sixth Painlevé equation derived

Differentials of elliptic modular functions presented

Abstract

We relate two parameter solutions of the sixth Painlev\'e equation and finite-gap solutions of the Heun equation by considering monodromy on a certain class of Fuchsian differential equations. In the appendix, we present formulae on differentials of elliptic modular functions, and obtain the ellitic form of the sixth Painlev\'e equation directly.