Combinatorial multiple Eisenstein series

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. We call these q-series combinatorial (bi-)multiple Eisenstein series, and in depth one they coincide with (classical) Eisenstein series. Combinatorial multiple Eisenstein series can be seen as an interpolation between the given mathbb {Q}-valued solution of the extended double shuffle equations (as qrightarrow 0) and multiple zeta values (as qrightarrow 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.