{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:TYSSCWENRQPPZGOFPHCDZAO7VK","short_pith_number":"pith:TYSSCWEN","schema_version":"1.0","canonical_sha256":"9e2521588d8c1efc99c579c43c81dfaa8168e7c96ff3b1c54471f7335eb81ecf","source":{"kind":"arxiv","id":"1409.7559","version":1},"attestation_state":"computed","paper":{"title":"Evaluation of Matrix-variate Gamma and Beta Integrals as Multiple Integrals and Kober Fractional Integral Operators in the Complex Matrix Variate Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"A.M. Mathai","submitted_at":"2014-09-26T12:50:25Z","abstract_excerpt":"Explicit evaluations of matrix-variate gamma and beta integrals in the complex domain by using conventional procedures is extremely difficult. Such an evaluation will reveal the structure of these matrix-variate integrals. In this article, explicit evaluations of matrix-variate gamma and beta integrals in the complex domain for the order of the matrix p = 1; 2 are given. Then fractional integral operators of the Kober type are given for some specific cases of the arbitrary function. A formal definition of fractional integrals in the complex matrix-variate case was given by the author earlier a"},"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":"1409.7559","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-26T12:50:25Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"57e5669a8853c185378a1a7236228d21b163805a74e8d7de9f8eda3fba6e6b65","abstract_canon_sha256":"85c6a7d5de8492c77698c2a2be01fc007ae1a7a9bd270985c16469550081feca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:52.426034Z","signature_b64":"iDVX7sqYJ6C+BpNp7AUzkZDpJQkM8XjXyTty6mRzgWHOD8Jfjpbw+uOXwymvMSGhxqoS1+U3BuxZuz2zGBQnCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e2521588d8c1efc99c579c43c81dfaa8168e7c96ff3b1c54471f7335eb81ecf","last_reissued_at":"2026-05-18T02:41:52.425594Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:52.425594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Evaluation of Matrix-variate Gamma and Beta Integrals as Multiple Integrals and Kober Fractional Integral Operators in the Complex Matrix Variate Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"A.M. Mathai","submitted_at":"2014-09-26T12:50:25Z","abstract_excerpt":"Explicit evaluations of matrix-variate gamma and beta integrals in the complex domain by using conventional procedures is extremely difficult. Such an evaluation will reveal the structure of these matrix-variate integrals. In this article, explicit evaluations of matrix-variate gamma and beta integrals in the complex domain for the order of the matrix p = 1; 2 are given. Then fractional integral operators of the Kober type are given for some specific cases of the arbitrary function. A formal definition of fractional integrals in the complex matrix-variate case was given by the author earlier a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7559","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":"1409.7559","created_at":"2026-05-18T02:41:52.425654+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.7559v1","created_at":"2026-05-18T02:41:52.425654+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.7559","created_at":"2026-05-18T02:41:52.425654+00:00"},{"alias_kind":"pith_short_12","alias_value":"TYSSCWENRQPP","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_16","alias_value":"TYSSCWENRQPPZGOF","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_8","alias_value":"TYSSCWEN","created_at":"2026-05-18T12:28:52.271510+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/TYSSCWENRQPPZGOFPHCDZAO7VK","json":"https://pith.science/pith/TYSSCWENRQPPZGOFPHCDZAO7VK.json","graph_json":"https://pith.science/api/pith-number/TYSSCWENRQPPZGOFPHCDZAO7VK/graph.json","events_json":"https://pith.science/api/pith-number/TYSSCWENRQPPZGOFPHCDZAO7VK/events.json","paper":"https://pith.science/paper/TYSSCWEN"},"agent_actions":{"view_html":"https://pith.science/pith/TYSSCWENRQPPZGOFPHCDZAO7VK","download_json":"https://pith.science/pith/TYSSCWENRQPPZGOFPHCDZAO7VK.json","view_paper":"https://pith.science/paper/TYSSCWEN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.7559&json=true","fetch_graph":"https://pith.science/api/pith-number/TYSSCWENRQPPZGOFPHCDZAO7VK/graph.json","fetch_events":"https://pith.science/api/pith-number/TYSSCWENRQPPZGOFPHCDZAO7VK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TYSSCWENRQPPZGOFPHCDZAO7VK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TYSSCWENRQPPZGOFPHCDZAO7VK/action/storage_attestation","attest_author":"https://pith.science/pith/TYSSCWENRQPPZGOFPHCDZAO7VK/action/author_attestation","sign_citation":"https://pith.science/pith/TYSSCWENRQPPZGOFPHCDZAO7VK/action/citation_signature","submit_replication":"https://pith.science/pith/TYSSCWENRQPPZGOFPHCDZAO7VK/action/replication_record"}},"created_at":"2026-05-18T02:41:52.425654+00:00","updated_at":"2026-05-18T02:41:52.425654+00:00"}