A q-analog of the hyperharmonic numbers

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


Recently, the q-analog of the harmonic numbers obtained by replacing each positive integer n with nq 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).

Original languageEnglish
Pages (from-to)147-160
Number of pages14
JournalAfrika Matematika
Issue number1
StatePublished - Mar 2014


  • Harmonic number
  • Hyperharmonic number
  • q-Generalization

ASJC Scopus subject areas

  • General Mathematics


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