On generalized hyperharmonic numbers of order r, H_{n,m}^{r} (\sigma)

Abstract

In this paper, we define generalized hyperharmonic numbers of order $r, H_{n,m}^{r}\left( \sigma \right) ,$ for $m\in \mathbb{Z}^{+}$ and give some applications by using generating functions of these numbers. For example, for $n, r, s\in \mathbb{Z}^{+}$ such that $1\leq s\leq r,$ \begin{equation*} \sum\limits_{k=1}^{n}\binom{n-k+s-1}{s-1}H_{k,m}^{r-s}\left( \sigma \right) =H_{n,m}^{r}\left( \sigma \right), \end{equation*} and \begin{equation*} \sum_{k=1}^{n}\sum_{i=1}^{k}\frac{H_{k-i,m}^{r+1}\left( \sigma \right) D_{r}(k-i+r)}{(n-k)!\left( k-i+r\right) !}=H_{n,m}^{2r+2}(\sigma ), \end{equation*} where $D_{r}(n)$ is an $r$-derangement number.