{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/square-series-generating-function","title":"Square Series Generating Function Transformations","arxiv_id":"1609.02803","date":"2016-09-09","proceeding":null,"authors":["Maxie D. Schmidt"],"abstract":"We construct new integral representations for transformations of the ordinary generating function for a sequence, $\\langle f_n \\rangle$, into the form of a generating function that enumerates the corresponding \"square series\" generating function for the sequence, $\\langle q^{n^2} f_n \\rangle$, at an initially fixed non-zero $q \\in \\mathbb{C}$. The new results proved in the article are given by integral-based transformations of ordinary generating function series expanded in terms of the Stirling numbers of the second kind. We then employ known integral representations for the gamma and double factorial functions in the construction of these square series transformation integrals. The results proved in the article lead to new applications and integral representations for special function series, sequence generating functions, and other related applications. A summary Mathematica notebook providing derivations of key results and applications to specific series is provided online as a supplemental reference to readers.","url_abs":"http://arxiv.org/abs/1609.02803v2","url_pdf":"http://arxiv.org/pdf/1609.02803v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"square-series-generating-function","repo_url":"https://github.com/maxieds/ManuscriptComputationalData","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}