On Ramanujan relations between Eisenstein series

The Ramanujan relations between Eisenstein series can be interpreted as an ordinary differential equation in a parameter space of a family of elliptic curves. Such an ordinary differential equation is inverse to the Gauss–Manin connection of the corresponding period map constructed by elliptic integrals of first and second kind. In this article we consider a slight modification of elliptic integrals by allowing non-algebraic integrands and we get in a natural way generalizations of Ramanujan relations between Eisenstein series.