Parameterizations of the Chazy equation

Abstract

The Chazy equation $y^{\prime \prime \prime }=2y{y}^{\prime \prime }-3y^{\prime 2}$ is derived from the automorphic properties of Schwarz triangle functions $S(\alpha ,\beta ,\gamma ;z)$. It is shown that solutions $y$ which are analytic in the fundamental domain of these triangle functions, only correspond to certain values of $\alpha ,\beta ,\gamma$. The solutions are then systematically constructed. These analytic solutions provide all known and one new parametrization of the Eisenstein series $P,Q,R$ introduced by Ramanujan in his modular theories of signature 2, 3, 4 and 6.

to appear in Studies in Applied Mathematics

References in corpus (2)

• On Rationally Parametrized Modular Equations
• Integrable Lagrangians and modular forms