{"paper":{"title":"On a $_2F_1\\big(\\frac{1}{4}\\big)$-identity due to Gosper","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Integration on a Gosper 2F1 identity produces a gamma closed form for a hypergeometric series at a large rational argument.","cross_cats":[],"primary_cat":"math.CA","authors_text":"Cetin Hakimoglu-Brown","submitted_at":"2026-04-06T16:05:46Z","abstract_excerpt":"It is only in exceptional cases that a $_2F_1(z)$-series with rational parameters and a rational argument, apart from the cases for $z \\in \\{ \\pm 1, \\frac{1}{2} \\}$ associated with classical hypergeometric identities, admits an evaluation given by a combination of $\\Gamma$-values with rational arguments. In this paper, we present a new and integration-based approach toward the construction of special values for $_2F_1$-series of the desired form. We apply this approach using a $_2F_1\\big(\\frac{1}{4}\\big)$-identity originally due to Gosper and later considered by Vidunas, Ebisu, and Zudilin, to"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We present a new and integration-based approach toward the construction of special values for 2F1-series of the desired form. We apply this approach using a 2F1(1/4)-identity originally due to Gosper ... to evaluate a 2F1-series of convergence rate (172872/185039)^2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the proposed integration-based construction actually produces a valid closed-form evaluation in terms of gamma values for the specific series considered, without hidden assumptions about convergence or analytic continuation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A new integration approach is used to evaluate a 2F1 series at argument (172872/185039)^2, extending a Gosper identity and claiming the largest numerator/denominator among known strange evaluations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Integration on a Gosper 2F1 identity produces a gamma closed form for a hypergeometric series at a large rational argument.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4cf707f4c78f36b4cada638c271270fa18298a81da1a47289655736996ba54cb"},"source":{"id":"2604.04799","kind":"arxiv","version":2},"verdict":{"id":"b65c2e57-3c5d-499c-b0cf-349653598cb8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T19:00:47.490359Z","strongest_claim":"We present a new and integration-based approach toward the construction of special values for 2F1-series of the desired form. We apply this approach using a 2F1(1/4)-identity originally due to Gosper ... to evaluate a 2F1-series of convergence rate (172872/185039)^2.","one_line_summary":"A new integration approach is used to evaluate a 2F1 series at argument (172872/185039)^2, extending a Gosper identity and claiming the largest numerator/denominator among known strange evaluations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the proposed integration-based construction actually produces a valid closed-form evaluation in terms of gamma values for the specific series considered, without hidden assumptions about convergence or analytic continuation.","pith_extraction_headline":"Integration on a Gosper 2F1 identity produces a gamma closed form for a hypergeometric series at a large rational argument."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.04799/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":1,"snapshot_sha256":"ebda02f366c92e2714b9bc9733c30d23d60d864917f68df73a135c0573f0223d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}