{"paper":{"title":"Some new series for $1/\\pi$ motivated by congruences","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":"2020-09-09T15:58:08Z","abstract_excerpt":"In this paper, we deduce a family of six new series for $1/\\pi$; for example, $$\\sum_{n=0}^\\infty\\frac{41673840n+4777111}{5780^n}W_n\\left(\\frac{1444}{1445}\\right) =\\frac{147758475}{\\sqrt{95}\\,\\pi}$$ where $W_n(x)=\\sum_{k=0}^n\\binom nk\\binom{n+k}k\\binom{2k}k\\binom{2(n-k)}{n-k}x^k$. To do so, we transform our series to series of the type $$\\sum_{n=0}^\\infty\\frac{an+b}{m^n}\\sum_{k=0}^n\\binom nk^4$$ studied by Cooper in 2012. In addition, we pose $17$ new series for $1/\\pi$ motivated by congruences; for example, we conjecture that $$\\sum_{k=0}^\\infty\\frac{4290k+367}{3136^k}\\binom{2k}kT_k(14,1)T_k("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.04379","kind":"arxiv","version":4},"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/2009.04379/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"}