The algebra of bi-brackets and regularised multiple Eisenstein series

TL;DR

This paper explores the algebraic structure of bi-brackets, a class of $q$-series related to modular forms and multiple zeta values, and introduces regularized multiple Eisenstein series satisfying shuffle and stuffle product relations.

Contribution

The paper develops a unified algebraic framework for bi-brackets and regularized multiple Eisenstein series, connecting them through shuffle and stuffle products, and derives new double shuffle relations.

Findings

Bi-brackets generalize multiple divisor sum generating functions.

Defined two types of regularized Eisenstein series, $G^{sh}$ and $G^*$, satisfying shuffle and stuffle relations.

Established new finite double shuffle relations for multiple Eisenstein series.

Abstract

We study the algebra of certain $q$-series, called bi-brackets, whose coefficients are given by weighted sums over partitions. These series incorporate the theory of modular forms for the full modular group as well as the theory of multiple zeta values (MZV) due to their appearance in the Fourier expansion of regularised multiple Eisenstein series. Using the conjugation of partitions we obtain linear relations between bi-brackets, called the partition relations, which yield naturally two different ways of expressing the product of two bi-brackets similar to the stuffle and shuffle product of multiple zeta values. Bi-brackets are generalizations of the generating functions of multiple divisor sums, called brackets, $[s_1,\dots,s_l]$ studied by the author and U. K\"uhn. We use the algebraic structure of bi-brackets to define further $q$-series $[s_1,\dots,s_l]^{sh}$ and $[s_1,\dots,s_l]^\ast$ which satisfy the shuffle and stuffle product formulas of MZV by using results about quasi-shuffle algebras introduced by Hoffman. In a recent work the author and K. Tasaka defined regularised multiple Eisenstein series $G^{sh}$, by using an explicit connection to the coproduct on formal iterated integrals. These satisfy the shuffle product formula. Applying the same concept for the coproduct on quasi-shuffle algebras enables us to define multiple Eisenstein series $G^\ast$ satisfying the stuffle product. We show that both $G^{sh}$ and $G^\ast$ are given by linear combinations of products of MZV and bi-brackets. Comparing these two regularized multiple Eisenstein series enables us to obtain finite double shuffle relations for multiple Eisenstein series in low weights which extend the relations proven before.