The algebra of bi-brackets and regularized multiple Eisenstein series

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 regularized 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. In a recent work, the author and K. Tasaka defined (shuffle) regularized multiple Eisenstein series G(E), by using an explicit connection to the coproduct on formal iterated integrals introduced by Goncharov. These satisfy the shuffle product formula. Applying the same concept for the coproduct on quasi-shuffle algebras enables us to define (stuffle) regularized multiple Eisenstein series G* satisfying the stuffle product formula. We show that both G(W) and G* are given by linear combinations of products of MZV and bi-brackets. (C) 2019 Elsevier Inc. All rights reserved.