{"paper":{"title":"The Graham--Knuth--Patashnik recurrence: Symmetries and continued fractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan D. Sokal, Jes\\'us Salas","submitted_at":"2020-08-07T10:07:30Z","abstract_excerpt":"We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (\\alpha n + \\beta k + \\gamma)\\, T(n-1,k)+(\\alpha' n + \\beta' k + \\gamma') \\, T(n-1,k-1)$ with initial condition $T(0,k) = \\delta_{k0}$ and parameters $\\mathbf{\\mu} = (\\alpha,\\beta,\\gamma, \\alpha',\\beta',\\gamma')$. We show that the family of arrays $T(\\mathbf{\\mu})$ is invariant under a 48-element discrete group isomorphic to $S_3 \\times D_4$. Our main result is to determine all parameter sets $\\mathbf{\\mu} \\in \\mathbb{C}^6$ for which the ordinary generating function $f(x,t) = \\sum_{n,k=0}^\\infty T(n,k) \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2008.03070","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2008.03070/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}