[2205.00879] An invitation to formal power series Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > History and Overview arXiv:2205.00879 (math) [Submitted on 28 Apr 2022 ( v1 ), last revised 1 Aug 2026 (this version, v7)] Title: An invitation to formal power series Authors: Benjamin Sambale View a PDF of the paper titled An invitation to formal power series, by Benjamin Sambale View PDF HTML (experimental) Abstract: This is a lecture on the theory of formal power series developed entirely without any analytic machinery. Combining ideas from various authors we are able to prove Newton's binomial theorem, Jacobi's triple product, the Rogers--Ramanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujan's partition congruences, generating functions of Stirling numbers and Jacobi's four-square theorem. We further discuss formal Laurent series and multivariate power series and end with a proof of MacMahon's master theorem. Comments: 5th version: Many corrections by Darij Grinberg, thanks! 6th+7th version: Typos corrected with Claude Opus Subjects: History and Overview (math.HO) ; Combinatorics (math.CO); Number Theory (math.NT) MSC classes: 13F25, 16W60, 11D88, 11P84, 05A15, 05A17 Cite as: arXiv:2205.00879 [math.HO] (or arXiv:2205.00879v7 [math.HO] for this version) https://doi.org/10.48550/arXiv.2205.00879 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Jahresber. Dtsch. Math.-Ver. 125 (2023), 3--69 Submission history From: Benjamin Sambale [ view email ] [v1] Thu, 28 Apr 2022 15:07:34 UTC (48 KB) [v2] Thu, 18 Aug 2022 09:33:45 UTC (48 KB) [v3] Sun, 8 Jan 2023 05:21:32 UTC (48 KB) [v4] Mon, 21 Aug 2023 18:36:20 UTC (50 KB) [v5] Sat, 16 Mar 2024 16:34:12 UTC (62 KB) [v6] Sun, 26 Apr 2026 05:53:04 UTC (63 KB) [v7] Sat, 1 Aug 2026 13:32:35 UTC (63 KB) Full-text links: Access Paper: View a PDF of the paper titled An invitation to formal power series, by Benjamin Sambale View PDF HTML (experimental) TeX Source view license Current browse context: math.HO < prev | next > new | recent | 2022-05 Change to browse by: math math.CO math.NT 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