Realizations of the formal double Eisenstein space

Abstract

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 ${\mathcal{E}}_k\to A$, 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 ${\rho }^{\mathfrak{K}}:{\mathcal{E}}_k\to \mathbb{Q}[[q]]$, which lifts Gangl-Kaneko-Zagier's Bernoulli realization ${\rho }^B:{\mathcal{D}}_k\to \mathbb{Q}$, 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.

16 pages, comments welcome! (V2: Typos corrected in Prop. 2.5,2.7 and 4.1)

References in corpus (1)

• q-analogues of multiple zeta values and the formal double Eisenstein space