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Euler Sums of Hyperharmonic Numbers

Year: 
2012
Researcher(s): 
Ayhan Dil, Khristo Boyadzhiev
Institution: 
Ohio Northern University
Discipline: 
Mathematics

The hyperharmonic numbers h_{n}^{(r)} are defined by means of the clasical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: {\sigma}(r,m)=\sum_{n=1}^{\infty}((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 Mezo and Dil. We also provide an explicit evaluation of {\sigma}(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.

ArXiV link - http://arxiv.org/abs/1209.0604

ArXiV e-print - Number Theory - http://arxiv.org/pdf/1209.0604v1.pdf