On Evaluations of Euler-Type Sums of Hyperharmonic Numbers

We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers hnr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{n}^{\left( r\right) }$$\end{document} with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of hyperharmonic numbers to an arbitrary integer r. Moreover, we reach at explicit formulas for the shifted Euler-type sums of harmonic and hyperharmonic numbers. All the evaluations are provided in terms of the Riemann zeta values, harmonic numbers and linear Euler sums.