标题： 欧拉和的广义超调和数 显示英文标题

标题： Euler sums of generalized hyperharmonic numbers

摘要： 广义超调和数$h_n^{(m)}(k)$是通过多重调和数定义的。 我们证明了超调和数$h_n^{(m)}(k)$满足某种递推关系，这使我们能够用经典的调和数来表示它们。 此外，我们证明了带有超调和数的欧拉型求和：\[S\left( {k,m;p} \right): = \sum\limits_{n = 1}^\infty {\frac{{h_n^{\left( m \right)}\left( k \right)}}{{{n^p}}}} \;\;\left(p\geq m+1,\ {k = 1,2,3} \right)\]可以表示为黎曼zeta值和调和数乘积的有理线性组合。 这是Dil（2015）\cite{AD2015}和 Mez$\ddot{o}$（2010）\cite{M2010}结果的扩展。 考虑了一些有趣的新的结论和示例。 显示英文摘要

摘要： 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 (2015) \cite{AD2015} and Mez$\ddot{o}$ (2010) \cite{M2010}. Some interesting new consequences and illustrative examples are considered.