Euler Sums of Hyperharmonic Numbers

TL;DR

This paper investigates Euler-type sums involving hyperharmonic numbers, expressing them in terms of Hurwitz zeta functions and providing explicit closed-form evaluations using zeta values and Stirling numbers.

Contribution

It generalizes previous results by expressing hyperharmonic sums in terms of Hurwitz zeta functions and derives explicit formulas involving zeta values and Stirling numbers.

Findings

Expressed hyperharmonic sums in terms of Hurwitz zeta functions

Provided explicit closed-form evaluations using zeta values and Stirling numbers

Evaluated several series involving hyperharmonic numbers

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: {\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 Mez\H{o} 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.