{"paper":{"title":"Evaluations of some series of the type $\\sum_{k=0}^\\infty(ak+b)x^k/\\binom{mk}{nk}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2022-04-11T16:39:25Z","abstract_excerpt":"In this paper, via the beta function we evaluate some series of the type $\\sum_{k=0}^\\infty(ak+b)x^k/\\binom{mk}{nk}$. For example, we prove that $$\\sum_{k=0}^\\infty\\frac{(49k+1)8^k}{3^k\\binom{3k}k}=81+16\\sqrt3\\,\\pi \\ \\ \\text{and}\\ \\ \\sum_{k=0}^\\infty\\frac{10k-1}{\\binom{4k}{2k}}=\\frac{4\\sqrt 3}{27}\\pi.$$\n  We also establish the following efficient formula for computing $\\log n$ with $1<n\\le 85/4$: \\begin{align*} &\\sum_{k=0}^\\infty\\frac{(2(n^2+6n+1)^2(n^2-10n+1)k+P(n))(n-1)^{4k}} {(-n)^k(n+1)^{2k}\\binom{4k}{2k}}\\\\ \\ \\ &=6n(n+1)(n-1)^3\\log n-32n(n+1)^2(n^2-4n+1), \\end{align*} where $$P(n):=n^6-58"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.08275","kind":"arxiv","version":7},"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/2204.08275/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"}