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

摘要

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