Combinatorial multiple Eisenstein series

TL;DR

The paper introduces combinatorial multiple Eisenstein series as $q$-series interpolating between solutions of double shuffle equations and multiple zeta values, linking modular forms and $q$-analogues.

Contribution

It constructs a new family of $q$-series satisfying extended double shuffle equations, generalizing multiple Eisenstein series and connecting them to multiple zeta values.

Findings

$q$-series satisfy extended double shuffle equations

They interpolate between solutions of shuffle equations and multiple zeta values

Explicit construction uses symmetril and swap invariant bimoulds

Abstract

We construct a family of $q$-series with rational coefficients satisfying a variant of the extended double shuffle equations, which are a lift of a given $\mathbb{Q}$-valued solution of the extended double shuffle equations. These $q$-series will be called combinatorial (bi-)multiple Eisenstein series, and in depth one they are given by Eisenstein series. The combinatorial multiple Eisenstein series can be seen as an interpolation between the given $\mathbb{Q}$-valued solution of the extended double shuffle equations (as $q\rightarrow 0$) and multiple zeta values (as $q\rightarrow 1$). In particular, they are $q$-analogues of multiple zeta values closely related to modular forms. Their definition is inspired by the Fourier expansion of multiple Eisenstein series introduced by Gangl-Kaneko-Zagier. Our explicit construction is done on the level of their generating series, which we show to be a so-called symmetril and swap invariant bimould.