Partitions, Multiple Zeta Values and the q-bracket

TL;DR

This paper develops a framework linking partition-based q-series to multiple zeta values, enabling the derivation of new relations and extending known results in the theory of q-analogues and multiple zeta values.

Contribution

It introduces a space of polynomial functions on partitions that serve as q-analogues of multiple zeta values and relates these to known structures like shifted symmetric functions.

Findings

Explicit description of regularized multiple zeta values as q→1

Relations among q-analogues of multiple zeta values derived

Functions on partitions yield quasimodular q-series

Abstract

We provide a framework for relating certain q-series defined by sums over partitions to multiple zeta values. In particular, we introduce a space of polynomial functions on partitions for which the associated q-series are q-analogues of multiple zeta values. By explicitly describing the (regularized) multiple zeta values one obtains as $q\to 1$, we extend previous results known in this area. Using this together with the fact that other families of functions on partitions, such as shifted symmetric functions, are elements in our space will then give relations among (q-analogues of) multiple zeta values. Conversely, we will show that relations among multiple zeta values can be `lifted' to the world of functions on partitions, which provides new examples of functions where the associated q-series are quasimodular.