Heun equation and Painlevé equation

Abstract

We relate two parameter solutions of the sixth Painlevé 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 Painlevé equation directly.

17 pages, based on the talk given at the Workshop "Studies on elliptic integrable systems" (Kyoto, November 2004)

References in corpus (1)

• Nodal curves and Riccati solutions of Painlevé equations

Cited by in corpus (5)

• Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations
• Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case
• Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
• Integral representation of solutions to Fuchsian system and Heun's equation
• The Elliptic Painlevé Lax Equation vs. van Diejen's 8-Coupling Elliptic Hamiltonian