Abstract
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, which reduces to the aforementioned $$q$$ -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 $$1,2,\ldots ,r$$ belong to distinct cycles (which is enumerated by the $$r$$ -Stirling number of the first kind).
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