Integral representation for Euler sums of hyperharmonic numbers

2020/07/03 by Casimir Rönnlöf, Rönnlöf, Casimir
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2007.01894

openalex publication_date 2020/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∫₀^∞ \fract^m-1ln(1-e^-t)(1-e^-t)^r dt where m, r ∈ ℕ, m > r and r≥1.

Citations

• Summation of Hyperharmonic Series

Related

• Euler Sums of Hyperharmonic Numbers
• Euler sums of generalized hyperharmonic numbers
• Computation and theory of Euler sums of generalized hyperharmonic numbers
• On Evaluations of Euler-type Sums of Hyperharmonic Numbers
• Euler sums of generalized hyperharmonic numbers