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The asymptotic behaviour of $B(x,y;p)$ is obtained when (i) $p$ large, with $x$ and $y$ fixed, (ii) $x$ and $p$ large, (iii) $x$, $y$ and $p$ large and (iv) either $x$ or $y$ large, with $p$ finite. Numerical results are given to illustrate the accuracy of the formulas obtained."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1503.04016","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-03-13T10:49:27Z","cross_cats_sorted":[],"title_canon_sha256":"c9470094f70e27a0d600f372119de608b12834f211505c5b2f79cf5000dfc391","abstract_canon_sha256":"b48bf6ac9b30cc4026ec185c261c724a95db8315bf4938b58c0651f924506c8b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:23:57.373477Z","signature_b64":"rmEP9pC/seTMblhDjkUH+LZghxLGSlXAiwQcM1QZmoV8dWTwfxC9PxaBfms8Upk2a4cstPsDSXN0iGwY4H4bBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9306f096e7866e309051ea7f52942564b915c00f74d9a78a31f8f212beab207a","last_reissued_at":"2026-05-18T02:23:57.372756Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:23:57.372756Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The asymptotics of a generalised Beta function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"R. B. Paris","submitted_at":"2015-03-13T10:49:27Z","abstract_excerpt":"We consider the generalised Beta function introduced by Chaudhry {\\it et al.\\/} [J. Comp. Appl. Math. {\\bf 78} (1997) 19--32] defined by \\[B(x,y;p)=\\int_0^1 t^{x-1} (1-t)^{y-1} \\exp \\left[\\frac{-p}{4t(1-t)}\\right]\\,dt,\\] where $\\Re (p)>0$ and the parameters $x$ and $y$ are arbitrary complex numbers. The asymptotic behaviour of $B(x,y;p)$ is obtained when (i) $p$ large, with $x$ and $y$ fixed, (ii) $x$ and $p$ large, (iii) $x$, $y$ and $p$ large and (iv) either $x$ or $y$ large, with $p$ finite. Numerical results are given to illustrate the accuracy of the formulas obtained."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.04016","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1503.04016","created_at":"2026-05-18T02:23:57.372874+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.04016v1","created_at":"2026-05-18T02:23:57.372874+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.04016","created_at":"2026-05-18T02:23:57.372874+00:00"},{"alias_kind":"pith_short_12","alias_value":"SMDPBFXHQZXD","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"SMDPBFXHQZXDBECR","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"SMDPBFXH","created_at":"2026-05-18T12:29:42.218222+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SMDPBFXHQZXDBECR5J7VFFBFMS","json":"https://pith.science/pith/SMDPBFXHQZXDBECR5J7VFFBFMS.json","graph_json":"https://pith.science/api/pith-number/SMDPBFXHQZXDBECR5J7VFFBFMS/graph.json","events_json":"https://pith.science/api/pith-number/SMDPBFXHQZXDBECR5J7VFFBFMS/events.json","paper":"https://pith.science/paper/SMDPBFXH"},"agent_actions":{"view_html":"https://pith.science/pith/SMDPBFXHQZXDBECR5J7VFFBFMS","download_json":"https://pith.science/pith/SMDPBFXHQZXDBECR5J7VFFBFMS.json","view_paper":"https://pith.science/paper/SMDPBFXH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.04016&json=true","fetch_graph":"https://pith.science/api/pith-number/SMDPBFXHQZXDBECR5J7VFFBFMS/graph.json","fetch_events":"https://pith.science/api/pith-number/SMDPBFXHQZXDBECR5J7VFFBFMS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SMDPBFXHQZXDBECR5J7VFFBFMS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SMDPBFXHQZXDBECR5J7VFFBFMS/action/storage_attestation","attest_author":"https://pith.science/pith/SMDPBFXHQZXDBECR5J7VFFBFMS/action/author_attestation","sign_citation":"https://pith.science/pith/SMDPBFXHQZXDBECR5J7VFFBFMS/action/citation_signature","submit_replication":"https://pith.science/pith/SMDPBFXHQZXDBECR5J7VFFBFMS/action/replication_record"}},"created_at":"2026-05-18T02:23:57.372874+00:00","updated_at":"2026-05-18T02:23:57.372874+00:00"}