Abstract
The multiple zeta values (MZVs) possess a rich algebraic structure of algebraic relations, which is conjecturally determined by two different (shuffle and stuffle) products of a certain algebra of noncommutative words. In a recent work, Bachmann constructed a q-analogue of the MZVs—the so-called bi-brackets—for which the two products are dual to each other, in a very natural way. We overview Bachmann’s construction and discuss the radial asymptotics of the bi-brackets, its links to the MZVs, and related linear (in)dependence questions of the q-analogue.
Highlights
The multiple zeta values (MZVs) possess a rich algebraic structure of algebraic relations, which is conjecturally determined by two different products of a certain algebra of noncommutative words
One of the principal features of the latter q-MZVs is that they are well defined for any collection of integers s1, . . . , sl, so they do not require regularisation as the former q-MZVs and the Mathematics 2015, 3
The goal of this note is to make an algebraic setup for Bachmann’s double stuffle relations as well as to demonstrate that those relations reduce to the corresponding stuffle and shuffle relations in the limit as q → 1−
Summary
The following result allows one to control the asymptotic behaviour of the bi-brackets as q → 1− and as q approaches radially a root of unity. This produces an explicit version of the asymptotics used in [7] for proving some linear and algebraic results in the case l = 1. Note that the bijective correspondence between the bi-brackets and the zeta functions nr ds11 −1 · · · nrl l dsl l −1.
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