{"paper":{"title":"A Note on the 2F1 Hypergeometric Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.CA","authors_text":"Armen Bagdasaryan","submitted_at":"2009-12-04T19:04:19Z","abstract_excerpt":"The special case of the hypergeometric function $_{2}F_{1}$ represents the binomial series $(1+x)^{\\alpha}=\\sum_{n=0}^{\\infty}(\\:\\alpha n\\:)x^{n}$ that always converges when $|x|<1$. Convergence of the series at the endpoints, $x=\\pm 1$, depends on the values of $\\alpha$ and needs to be checked in every concrete case. In this note, using new approach, we reprove the convergence of the hypergeometric series $_{2}F_{1}(\\alpha,\\beta;\\beta;x)$ for $|x|<1$ and obtain new result on its convergence at point $x=-1$ for every integer $\\alpha\\neq 0$. The proof is within a new theoretical setting based o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.0917","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"}