[1810.07877] Generalized Harmonic Numbers Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:1810.07877 (math) [Submitted on 18 Oct 2018 ( v1 ), last revised 24 Apr 2026 (this version, v7)] Title: Generalized Harmonic Numbers Authors: Jose Risomar Sousa View a PDF of the paper titled Generalized Harmonic Numbers, by Jose Risomar Sousa View PDF HTML (experimental) Abstract: This paper presents new formulae for the harmonic numbers of order $k$, $H_{k}(n)$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)$ and $S^m_{k}(n)$. I believe this new formula for $H_{k}(n)$ is an improvement over the digamma function, $\psi$, because it's simpler and it stems from Faulhaber's formula, which provides a closed-form for the sum of powers of the first $n$ positive integers. We demonstrate how to create an exact power series for the harmonic numbers, a new integral representation for $\zeta(2k+1)$ and a new generating function for $\zeta(2k+1)$, among many other original results. The approaches and formulae discussed here are entirely different from solutions available in the literature. Comments: 35 pages. Set the grammar to academic standards and fixed cut-off formulas Subjects: Number Theory (math.NT) MSC classes: 11-XX Cite as: arXiv:1810.07877 [math.NT] (or arXiv:1810.07877v7 [math.NT] for this version) https://doi.org/10.48550/arXiv.1810.07877 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Jose Risomar Sousa Mr. [ view email ] [v1] Thu, 18 Oct 2018 02:30:59 UTC (24 KB) [v2] Sun, 21 Oct 2018 18:13:38 UTC (24 KB) [v3] Mon, 24 Jun 2019 03:52:47 UTC (15 KB) [v4] Tue, 25 Jun 2019 03:39:11 UTC (15 KB) [v5] Sat, 17 Oct 2020 22:08:00 UTC (15 KB) [v6] Sun, 31 Oct 2021 06:05:52 UTC (15 KB) [v7] Fri, 24 Apr 2026 18:44:47 UTC (16 KB) Full-text links: Access Paper: View a PDF of the paper titled Generalized Harmonic Numbers, by Jose Risomar Sousa View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2018-10 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from