On $3-2-1$ values of finite multiple harmonic $q$-series at roots of unity

TL;DR

This paper addresses open questions about finite multiple harmonic q-series at roots of unity, providing new results and conjectures that generalize previous findings in the field.

Contribution

It solves two open problems on 3-2-1 indices at roots of unity and introduces conjectures on cyclic sums that extend the current understanding.

Findings

Resolved two open questions about finite multiple harmonic q-series.

Proposed conjectures on cyclic sums generalizing existing results.

Enhanced understanding of q-series at roots of unity.

Abstract

We mainly answer two open questions about finite multiple harmonic $q$-series on 3-2-1 indices at roots of unity, posed recently by H. Bachmann, Y. Takeyama, and K. Tasaka. Two conjectures regarding cyclic sums which generalize the given results are also provided.