A q-analog of the hyperharmonic numbers

Recently, the q-analog of the harmonic numbers obtained by replacing each positive integer n with n_q has been shown to satisfy congruences which generalize Wolstenholme's theorem. Here, we wish to consider further algebraic properties of these numbers. Recall that the r-harmonic, or hyperharmonic, numbers arise by taking repeated partial sums of harmonic numbers. In this paper, we introduce and study properties of a q-analog of the r-harmonic numbers when r = 1. It is defined in terms of a statistic on the set of permutations of length n in which the elements 1,2,...,r belong to distinct cycles (which is enumerated by the r-Stirling number of the first kind).