A generalization of formal multiple zeta values related to multiple Eisenstein series and multiple q-zeta values

TL;DR

This paper introduces a new algebraic framework that unifies multiple Eisenstein series and multiple q-zeta values, establishing connections and embeddings between their formal structures and revealing their underlying relationships.

Contribution

It constructs the $ au$-invariant balanced quasi-shuffle algebra $ ext{G}^f$, formalizes multiple Eisenstein series and q-zeta values, and shows embeddings into existing algebraic schemes, advancing the understanding of their algebraic relations.

Findings

The algebra $ ext{G}^f$ formalizes multiple Eisenstein series and q-zeta values.

The affine scheme $ ext{BM}$ embeds Racinet's scheme $ ext{DM}$.

A projection from $ ext{G}^f$ to $ ext{Z}^f$ extracts constant terms or limits.

Abstract

We present the $\tau$-invariant balanced quasi-shuffle algebra $\mathcal{G}^{\operatorname{f}}$, whose elements formalize (combinatorial) multiple Eisenstein series as well as multiple q-zeta values. In particular, $\mathcal{G}^{\operatorname{f}}$ has natural maps into these two algebras, and we expect these maps to be isomorphisms. Racinet studied the algebra $\mathcal{Z}^f$ of formal multiple zeta values by examining the corresponding affine scheme DM. Similarly, we present the affine scheme BM corresponding to the algebra $\mathcal{G}^{\operatorname{f}}$. We show that Racinet's affine scheme DM embeds into our affine scheme BM. This leads to a projection from the algebra $\mathcal{G}^{\operatorname{f}}$ onto $\mathcal{Z}^f$. Via the above natural maps, this projection corresponds to extracting the constant terms of multiple Eisenstein series or the limit $q\to1$ of multiple q-zeta values.