Euler sums of hyperharmonic numbers

Abstract

The hyperharmonic numbers hn(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∞hn(r)nm can be expressed in terms of series of Hurwitz zeta function values. This is a generalization of a result of Mező and Dil (2010) [7]. 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.