Multiple $q$-zeta brackets

TL;DR

This paper explores the algebraic structure of multiple $q$-zeta brackets, a $q$-analogue of multiple zeta values, discussing their construction, asymptotics, and linear independence properties.

Contribution

It reviews Bachmann's construction of bi-brackets and analyzes their asymptotic behavior and algebraic relations, linking them to classical MZVs.

Findings

Bi-brackets' radial asymptotics relate to MZVs.

Duality of shuffle and stuffle products in bi-brackets.

Discussion on linear independence of $q$-analogues.

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.