Integral representation for Euler sums of hyperharmonic numbers

Abstract

In this short paper, we derive an integral representation for Euler sums of hyperharmonic numbers. We use results established by other authors to then show that the integral has a closed-form in terms of zeta values and Stirling numbers of the first kind. Specifically, the integral has the form of $$\int_0^\infty \frac{t^{m-1}\ln(1-e^{-t})}{(1-e^{-t})^r} \ dt$$ where $m, r \in \mathbb{N}$, $m > r$ and $r\ge1$.