Journal of the
Korean Mathematical Society JKMS

The generalized hyperharmonic numbers $h_n^{(m)}(k)$ are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers $h_n^{(m)}(k)$ satisfy certain recurrence relation which allow us to write them in terms of classical harmonic numbers. Moreover, we prove that the Euler-type sums with hyperharmonic numbers: \[S\left( {k,m;p} \right): = \sum\limits_{n = 1}^\infty {\frac{{h_n^{\left( m \right)}\left( k \right)}}{{{n^p}}}} \;\;\left(p\geq m+1,\ {k = 1,2,3} \right)\] can be expressed as a rational linear combination of products of Riemann zeta values and harmonic numbers. This is an extension of the results of Dil \cite{AD2015} and Mez\H{o} \cite{M2010}. Some interesting new consequences and illustrative examples are considered.

Keywords: Euler sums, generalized hyperharmonic numbers, harmonic numbers, Riemann zeta function, Stirling numbers

MSC numbers: 11B73, 11B83, 11M06, 11M32, 11M99