Integral representation for Euler sums of hyperharmonic numbers

TL;DR

This paper derives an integral representation for Euler sums involving hyperharmonic numbers and expresses it in closed form using zeta values and Stirling numbers, providing new insights into their structure.

Contribution

It introduces a novel integral representation for Euler sums of hyperharmonic numbers and expresses it in closed form with special functions.

Findings

Integral representation involving hyperharmonic numbers derived.

Closed-form expression in terms of zeta values and Stirling numbers.

Provides a new analytical tool for studying Euler sums.

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 $r\ge1$.