{"paper":{"title":"On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ${}_1F_0$ series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Hao Pan, Yong Zhang","submitted_at":"2018-10-22T15:38:35Z","abstract_excerpt":"Let $p$ be an odd prime and let $n$ be a positive integer. For any positive integer $\\alpha$ and $m\\in\\{1,2,3\\}$, we have \\begin{align*} \\sum_{k=0}^{p^{\\alpha}n-1}\\frac{(\\frac12)_k}{k!}\\cdot\\frac{(-4)^k}{m^k}\\equiv\\bigg(\\frac{m(m-4)}{p}\\bigg)\\sum_{k=0}^{p^{\\alpha-1}n-1}\\frac{(\\frac12)_k}{k!}\\cdot\\frac{(-4)^k}{m^k}\\pmod{p^{2\\alpha}}, \\end{align*} where $(x)_k=x(x+1)\\cdots(x+k-1)$ and $\\big(\\frac{\\cdot}{\\cdot}\\big)$ denotes the Legendre symbol. Also, when $m=4$, \\begin{align*} \\sum_{k=0}^{p^{\\alpha}n-1}(-1)^k\\cdot\\frac{(\\frac12)_k}{k!}\\equiv p\\sum_{k=0}^{p^{\\alpha-1}n-1}(-1)^k\\cdot\\frac{(\\frac12"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.09370","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":""},"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"}