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

TL;DR

This survey explores the algebraic structure of q-analogues of multiple zeta values, introduces the formal double Eisenstein space, and applies these concepts to derive combinatorial proofs of identities among modular forms.

Contribution

It introduces the formal double Eisenstein space and connects it with q-analogues of multiple zeta values, providing new algebraic insights and combinatorial proofs.

Findings

Algebraic structure of q-analogues of multiple zeta values elucidated

Formal double Eisenstein space defined and realized

Purely combinatorial proofs of modular form identities obtained

Abstract

In this survey article, we discuss the algebraic structure of q-analogues of multiple zeta values, which are closely related to derivatives of Eisenstein series. Moreover, we introduce the formal double Eisenstein space, which generalizes the formal double zeta space of Gangl, Kaneko, and Zagier. Using the algebraic structure of q-analogues of multiple zeta values, we will present a realization of this space. As an application, we will obtain purely combinatorial proofs of identities among (quasi-)modular forms.