{"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/the-factorial-basis-method-for-finding","title":"The Factorial-Basis Method for Finding Definite-Sum Solutions of Linear Recurrences With Polynomial Coefficients","arxiv_id":"2202.05550","date":"2022-02-11","proceeding":null,"authors":["Antonio Jiménez-Pastor","Marko Petkovšek"],"abstract":"The problem of finding a nonzero solution of a linear recurrence $Ly = 0$ with polynomial coefficients where $y$ has the form of a definite hypergeometric sum, related to the Inverse Creative Telescoping Problem of [14][Sec. 8], has now been open for three decades. Here we present an algorithm (implemented in a SageMath package) which, given such a recurrence and a quasi-triangular, shift-compatible factorial basis $\\mathcal{B} = \\langle P_k(n)\\rangle_{k=0}^\\infty$ of the polynomial space $\\mathbb{K}[n]$ over a field $\\mathbb{K}$ of characteristic zero, computes a recurrence satisfied by the coefficient sequence $c = \\langle c_k\\rangle_{k=0}^\\infty$ of the solution $y_n = \\sum_{k=0}^\\infty c_kP_k(n)$ (where, thanks to the quasi-triangularity of $\\mathcal{B}$, the sum on the right terminates for each $n \\in \\mathbb{N}$). More generally, if $\\mathcal{B}$ is $m$-sieved for some $m \\in \\mathbb{N}$, our algorithm computes a system of $m$ recurrences satisfied by the $m$-sections of the coefficient sequence $c$. If an explicit nonzero solution of this system can be found, we obtain an explicit nonzero solution of $Ly = 0$.","url_abs":"https://arxiv.org/abs/2202.05550v3","url_pdf":"https://arxiv.org/pdf/2202.05550v3.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":"the-factorial-basis-method-for-finding","repo_url":"https://github.com/antonio-jp/pseries_basis","is_official":1,"mentioned_in_paper":1,"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}