Balanced multiple q-zeta values

We introduce the balanced multiple q -zeta values. They give a new model for multiple q -zeta values, whose product formula combines the shuffle and stuffle product for multiple zeta values in a natural way. Moreover, the balanced multiple qzeta values are invariant under a very explicit involution. Thus, all relations among the balanced multiple q -zeta values are conjecturally of a very simple shape. Examples of the balanced multiple q -zeta values are the classical Eisenstein series, and they also contain the combinatorial multiple Eisenstein series introduced in [3]. The construction of the balanced multiple q -zeta values is done on the level of generating series. We introduce a general setup relating Hoffman's quasi -shuffle products to explicit symmetries among generating series of words, which gives a clarifying approach to Ecalle's theory of bimoulds. This allows us to obtain an isomorphism between the underlying Hopf algebras of words related to the combinatorial bi-multiple Eisenstein series and the balanced multiple q -zeta values. (c) 2024 Elsevier Inc. All rights reserved.