Ramanujan systems of Rankin-Cohen type and hyperbolic triangles

In the first part of the paper , we characterize certain systems of first-order nonlinear differential equations whose space of solutions is an sl(2)(C)-module. We prove that such systems, called Ramanujan systems of Rankin-Cohen type, have a special shape and are precisely the ones whose solution space admits a Rankin- Cohen structure. In the second part of the paper, we consider triangle groups ?(n, m, 8). By means of modular embeddings, we associate to every such group a number of systems of nonlinear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on ?(n, m, 8) are generated by the solutions of one such system (which is of Rankin-Cohen type). As a corollary , we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non-classical setting, we construct the space of integral weight twisted modular form on ?(2, 5, 00) from solutions of systems of nonlinear ODEs.