A note on the sum of finite multiple harmonic $q$-series on $r\text{-}(r+1)$ indices

TL;DR

This paper investigates finite multiple harmonic $q$-series at roots of unity for specific indices and provides conditions related to conjectures on cyclic sums of these series, advancing understanding in this mathematical area.

Contribution

It offers new insights into the sum of finite multiple harmonic $q$-series at roots of unity and establishes equivalent conditions for conjectures on cyclic sums.

Findings

Derived conditions equivalent to two recent conjectures.

Analyzed series at roots of unity for indices 1-3.

Enhanced understanding of cyclic sums in finite multiple harmonic $q$-series.

Abstract

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