Parameterizations of the Chazy equation
Abstract
The Chazy equation y''' = 2yy'' - 3y'2 is derived from the automorphic properties of Schwarz triangle functions S(α, β, γ; z). It is shown that solutions y which are analytic in the fundamental domain of these triangle functions, only correspond to certain values of α, β, γ. 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.
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