Duality and (q-)multiple zeta values

TL;DR

This paper explores duality relations in multiple zeta values and their q-analogs, revealing new connections and identities that deepen understanding of their algebraic structures.

Contribution

It extends Zudilin's duality framework to classical and q-analog multiple zeta values, identifying new relations and equivalences.

Findings

Derivation relation of order two matches Hoffman-Ohno type relation.

Relations between Ohno-Okuda-Zudilin and Schlesinger-Zudilin q-multiple zeta values.

Duality construction links classical and q-analog multiple zeta values.

Abstract

Following Bachmann's recent work on bi-brackets and multiple Eisenstein series, Zudilin introduced the notion of multiple q-zeta brackets, which provides a q-analog of multiple zeta values possessing both shuffle as well as quasi-shuffle relations. The corresponding products are related in terms of duality. In this work we study Zudilin's duality construction in the context of classical multiple zeta values as well as various q-analogs of multiple zeta values. Regarding the former we identify the derivation relation of order two with a Hoffman-Ohno type relation. Then we describe relations between the Ohno-Okuda-Zudilin q-multiple zeta values and the Schlesinger-Zudilin q-multiple zeta values.