Euler Sums of Hyperharmonic Numbers

2012/09/04 by Ayhan Dil, Dil, Ayhan, Khristo N. Boyadzhiev +1 · 1 citation
Mathematics · #11B73 #11M99 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT) #math.CA #math.NT #msc:11B73 #msc:11M99

paper · pdf · doi:10.48550/arxiv.1209.0604

9 pages

arxiv created 2013/11/05 · arxiv updated 2013/11/06

Abstract

The hyperharmonic numbers h_n^(r) are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: σ(r,m)=∑_n=1^∞((h_n^(r))/(n^m)) can be expressed in terms of series of Hurwitz zeta function values. This is a generalization of a result of Mező and Dil. We also provide an explicit evaluation of σ(r,m) in a closed form in terms of zeta values and Stirling numbers of the first kind. Furthermore, we evaluate several other series involving hyperharmonic numbers.

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• The Parity of Two Types of Cyclotomic Euler Sums via Contour Integrals

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