New series with Cauchy and Stirling numbers, Part 2

TL;DR

This paper derives new closed-form expressions for series involving Cauchy and Stirling numbers, including Euler sums of hyperharmonic numbers, expanding the understanding of these special number series.

Contribution

It provides novel closed-form evaluations of series with Cauchy and Stirling numbers, including new results for Euler sums involving hyperharmonic numbers.

Findings

Closed-form expressions for series with Cauchy and Stirling numbers.

New Euler sum formulas involving hyperharmonic numbers.

Enhanced understanding of special number series relationships.

Abstract

We evaluate in closed form several series involving products of Cauchy numbers with other special numbers (harmonic, skew-harmonic, hyperharmonic, and central binomial). Similar results are obtained with series involving Stirling numbers of the first kind. We focus on several particular cases which give new closed forms for Euler sums of hyperharmonic numbers and products of hyperharmonic and harmonic numbers.