{"paper":{"title":"Proof of conjectures on series with summands involving $ \\binom{2k}{k}8^k/(\\binom{3k}{k}\\binom{6k}{3k})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Yajun Zhou, Zhi-Wei Sun","submitted_at":"2024-01-25T14:18:15Z","abstract_excerpt":"Using cyclotomic multiple zeta values of level $8$, we confirm and generalize several conjectural identities on infinite series with summands involving $\\binom{2k}k8^k/(\\binom{3k}k\\binom{6k}{3k})$. For example, we prove that \\[\\sum_{k=0}^\\infty\\frac{(350k-17)\\binom{2k}k8^k} {\\binom{3k}k\\binom{6k}{3k}}=15\\sqrt2\\,\\pi+27\\] and \\[\\sum_{k=1}^\\infty\\frac{\\left\\{(5k-1)\\left[16\\mathsf H_{2k-1}^{(2)}-3\\mathsf H_{k-1}^{(2)}\\right]-\\frac{12(6k-1)}{(2k-1)^2}\\right\\}\\binom{2k}k8^k} {k(2k-1)\\binom{3k}k\\binom{6k}{3k}}=\\frac{\\pi^3}{12\\sqrt2},\\] where $\\mathsf H^{(2)}_m$ denotes the second-order harmonic numbe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.14197","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/2401.14197/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"}