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, thereby extending our earlier results on Hecke groups. 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 Eisenstein series associated to the triangle group and its orbit under the group action. We conclude by discussing the Painlevé property of these nonlinear differential equations.