{"paper":{"title":"Directional convexity and characterizations of Beta and Gamma functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Janu sz Matkowski, Martin Himmel","submitted_at":"2015-04-22T17:42:29Z","abstract_excerpt":"The logarithmic convexity of restrictions of the Beta functions to rays parallel to the main diagonal and the functional equation \\[ \\phi\\left( x+1\\right) =\\frac{x\\left( x+k\\right) }{\\left( 2x+k+1\\right) \\left( 2x+k\\right) }\\phi\\left( x\\right) ,\\ \\ \\ \\ \\ \\ x>0, \\] for $k>0$ allow to get a characterizations of the Beta function. This fact and a notion of the beta-type function lead to a new characterization of the Gamma function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01472","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"}