On 3-2-1 values of finite multiple harmonic \(q\)-series at roots of unity (Q2165748)

Recently, H. Bachmann, Y. Takeyama, and K. Tasaka studied special values of finite multiple harmonic \(q\)-series at roots of unity [\textit{H. Bachmann} et al., IRMA Lect. Math. Theor. Phys. 32, 1--18 (2020; Zbl 1439.11218)]. These objects were recently introduced by them and it was shown that they have connections to finite and symmetric multiple zeta values and the Kaneko-Zagier conjecture. Furthermore, they also formulated two open questions regarding explicit evaluations for finite multiple harmonic \(q\)-series on 3-2-1 indices at roots of unity. In the paper under review, the authors answer these questions affirmatively and they provide two conjectures regarding cyclic sums which generalize the given results.