Some $q$-congruences for homogeneous and quasi-homogeneous 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.