Triangle Groups: Automorphic Forms and Nonlinear Differential Equations.

We study the relations governing the ring of quasiautomorphic forms associated to triangle groups with a single cusp. The Eisenstein series associated to these triangle groups are shown to satisfy Ramanujan-like identities. These identities in turn allow us to associate a nonlinear differential equation to each triangle group. We show that they are solved by the quasiautomorphic weight-2 Eisenstein series associated to the triangle group and its orbit under the group action. We conclude by demonstrating that these nonlinear equations possess the Painleve property.