{"paper":{"title":"Log-concavity and log-convexity in the theory of the Graham--Knuth--Patashnik recurrences","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jes\\'us Salas","submitted_at":"2026-07-05T10:20:37Z","abstract_excerpt":"We study the triangular array $T(n,k;\\mu)$ defined by the Graham--Knuth--Patashnik recurrences $$ T(n,k) \\;=\\; (\\alpha n + \\beta k + \\gamma) \\, T(n-1,k) +\n  (\\alpha' n + \\beta' k + \\gamma') \\, T(n-1,k-1) $$ with initial condition $T(0,k)=\\delta_{k,0}$ and parameters $\\mu=(\\alpha,\\beta,\\gamma,\\alpha',\\beta',\\gamma')$, which are considered to be indeterminates. We first prove that, for any fixed $n\\ge 0$, the sequence $(T(n,k;\\mu))_{k\\ge 0}$ is strongly log-concave with the coefficientwise partial order in the variables $\\alpha,\\beta,\\gamma,\\alpha',\\beta',\\gamma'$. Moreover, we show that the seq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04217","kind":"arxiv","version":1},"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/2607.04217/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"}