Some $q$-congruences for homogeneous and quasi-homogeneous multiple $q$-harmonic sums

TL;DR

This paper establishes new $q$-congruences for multiple $q$-harmonic sums with specific index patterns, extending classical harmonic sum congruences to the $q$-analogue and proposing related conjectures.

Contribution

It introduces novel Wolstenholme type $q$-congruences for multiple $q$-harmonic sums, generalizing previous results to arbitrary depth and specific index strings.

Findings

New $q$-congruences for multiple $q$-harmonic sums

Extension of classical harmonic sum congruences to $q$-analogues

Proposed conjecture on cyclic sums of multiple $q$-harmonic sums

Abstract

We show some new Wolstenholme type $q$-congruences for some classes of multiple $q$-harmonic sums of arbitrary depth with strings of indices composed of ones, twos and threes. Most of these results are $q$-extensions of the corresponding congruences for ordinary multiple harmonic sums obtained by the authors in a previous paper. Finally, we pose a conjecture concerning two kinds of cyclic sums of multiple $q$-harmonic sums.