A note on the sum of finite multiple harmonic q-Series on r-(r + 1) indices

Abstract

We study the sum of the finite multiple harmonic q-series on r-(r + 1) indices at roots of unity with r = 1, 2, 3. Mainly, we give equivalent  conditions of two conjectures regarding cyclic sums of finite multiple harmonic q-series on 1-2-3 indices at roots of unity, posed recently  by Kh. Pilehrood, T. Pilehrood and R. Tauraso.