On the structure of Multiple q-Zeta Values

TL;DR

This paper investigates the structure of multiple q-zeta values, proving a conjecture for a specific subspace and proposing an explicit approach to the full conjecture using duality-based relations.

Contribution

It proves Bachmann's conjecture for a subspace of multiple q-zeta values and offers a method to approach the full conjecture through duality relations.

Findings

Confirmed the conjecture for a subspace of q-zeta values.

Provided an explicit approach to the full conjecture.

Identified relations among q-zeta values implied by duality.

Abstract

In 2015, Bachmann \cite{Ba3} conjectured that the~$\Q$-vector space~$\Zq$ of (formal)~$q$-analogues of Multiple Zeta Values (\qmzv s) is spanned by a very particular set compared to known spanning sets. In this work, we prove that this conjecture is true for a subspace of~$\Zq$ spanned by words satisfying some condition on their number of zeros and depth. According to this partial result, we give an explicit approach to the whole conjecture, based on particular~$\Q$-linear relations among formal Multiple~$q$-Zeta Values which are implied by duality.