Euler sums of generalized hyperharmonic numbers
arXiv:1701.00391
Abstract
The generalized hyperharmonic numbers are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers 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 (2015) \cite{AD2015} and Mez (2010) \cite{M2010}. Some interesting new consequences and illustrative examples are considered.