{"paper":{"title":"Weighted Derivative Sums of a Gamma Quotient: Sun's Conjecture and Cyclotomic Specializations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Shivam Nalin Patel","submitted_at":"2026-07-13T19:37:26Z","abstract_excerpt":"Let $f(x) = \\Gamma(x)^2/(2\\Gamma(2x))$ and set $\\lambda_\\alpha = 4\\sin^2\\alpha$ for $0 < \\alpha < \\pi/2$. We establish an elementary parameter identity for a weighted translate of $f$ and derive an explicit formula, valid at every derivative order, for the associated weighted sums $\\sum_{k\\ge1} \\lambda_\\alpha^{k-1} f^{(r)}(k)$. The coefficients satisfy an effective recurrence in ordinary zeta values. The unique unweighted specialization $\\alpha = \\pi/6$ proves Conjecture 4.1 of Zhi-Wei Sun; at the fourth order a depth-two value $\\mathrm{Gl}_{4,1}(\\pi/3)$ occurs. The construction complements ge"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15303","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/2607.15303/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"}