On quintic Eisenstein series and points of order five of the Weierstrass elliptic functions

We employ a new constructive approach to study modular forms of level five by evaluating the Weierstrass elliptic functions at points of order five on the period parallelogram. A significant tool in our analysis is a nonlinear system of coupled differential equations analogous to Ramanujan’s differential system for the Eisenstein series on SL(2,ℤ). The resulting relations of level five may be written as a coupled system of differential equations for quintic Eisenstein series. Some interesting combinatorial and analytic consequences result, including an alternative proof of a famous identity of Ramanujan involving the Rogers–Ramanujan continued fraction