Realizations of the formal double Eisenstein space

We introduce the formal double Eisenstein space $\mathcal{E}_k$, which is a generalization of the formal double zeta space $\mathcal{D}_k$ of Gangl-Kaneko-Zagier, and prove analogues of the sum formula and parity result for formal double Eisenstein series. We show that $\mathbb Q$-linear maps $\mathcal{E}_k\rightarrow A$, for some $\mathbb Q$-algebra $A$, can be constructed from formal Laurent series (with coefficients in $A$) that satisfy the Fay identity. As the prototypical example, we define the Kronecker realization $\rho^{\mathfrak{K}}: \mathcal{E}_k\rightarrow \mathbb Q[[q]]$, which lifts Gangl-Kaneko-Zagier's Bernoulli realization $\rho^B: \mathcal{D}_k\rightarrow \mathbb Q$, and whose image consists of quasimodular forms for the full modular group. As an application to the theory of modular forms, we obtain a purely combinatorial proof of Ramanujan's differential equations for classical Eisenstein series.