Proving dualities for qMZVs with connected sums

TL;DR

This paper introduces a novel approach using connected sums to prove dualities in qMZVs, simultaneously establishing dualities for multiple models and deriving important relations like the shuffle product and q-Ohno relations.

Contribution

It applies connected sums to prove dualities for Schlesinger-Zudilin and Bradley-Zhao models of qMZVs simultaneously, unifying several important results.

Findings

Duality for Schlesinger-Zudilin and Bradley-Zhao qMZV models proved.

Duality for MZVs derived from the qMZV dualities.

q-Ohno relations established as a generalization of Bradley-Zhao duality.

Abstract

This paper gives a new application of so-called connected sums, introduced recently by Seki and Yamamoto. Special about our approach is that it proves a duality for the Schlesinger-Zudilin and the Bradley-Zhao model of qMZVs simultaneously. The latter implies the duality for MZVs and the former can be used to prove the shuffle product formula for MZVs. Furthermore, the $q$-Ohno relations, a generalization of Bradley-Zhao duality, is also obtained.