Multiple zeta values and their q-analogues

TL;DR

This paper investigates multiple zeta values and their $q$-analogues, exploring their algebraic relations, generalizations, and a specific case of a conjecture about their generated spaces.

Contribution

It introduces a $q$-generalization of MZVs, analyzes their algebraic relations, and proves a particular case of Bachmann's conjecture on the equality of generated spaces.

Findings

Established a specific case of Bachmann's conjecture.

Demonstrated that $q$-analogues satisfy similar algebraic relations as classical MZVs.

Connected $q$-generalizations to underlying algebraic structures.

Abstract

We explore the theory of multiple zeta values (MZVs) and some of their $q$-generalisations. Multiple zeta values are numerical quantities that satisfy several combinatorial relations over the rationals. These relations include two multiplicative relations, which arise naturally from comparison of the MZVs with an underlying algebraic structure. We generalise these concepts by introducing the parameter $q$ in such a way that as $q\to 1^-$ we return to the ordinary MZVs. Our special interest lies in two $q$-models recently introduced by H. Bachmann. He further conjectures that the $\mathbb{Q}$-spaces generated by these $q$-generalisations coincide. In this thesis we establish a particular case of Bachmann's conjecture.