Maesaka-Seki-Watanabe's formula for multiple harmonic $q$-sums

TL;DR

This paper introduces a $q$-analogue of the MSW formula for multiple harmonic $q$-sums, providing new proofs and generalizations that deepen understanding of these series at roots of unity.

Contribution

It develops a $q$-analogue of the MSW formula, offers a new proof of the duality relation, and generalizes the formula for Schur type series.

Findings

Established a $q$-analogue of the MSW formula.

Provided a new proof of the duality relation at roots of unity.

Extended the formula to Schur type series.

Abstract

Maesaka, Seki, and Watanabe recently discovered an equality called the MSW formula. This paper provides a $q$-analogue of the MSW formula. It discusses the new proof of the duality relation for finite multiple harmonic $q$-series at primitive roots of unity via $q$-analogue of the MSW formula. This paper also gives a $q$-analogue of Yamamoto's generalization of the MSW formula for Schur type.