{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:TTYROSVXMHOZQOW4QL2ZAW6ONV","short_pith_number":"pith:TTYROSVX","schema_version":"1.0","canonical_sha256":"9cf1174ab761dd983adc82f5905bce6d42ea0ad534d1efd0b643b4c937d9a36d","source":{"kind":"arxiv","id":"1509.06465","version":1},"attestation_state":"computed","paper":{"title":"Some new properties of Confluent Hypergeometric Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CV","authors_text":"Wei-Chuan Lin, Xu-Dan Luo","submitted_at":"2015-09-22T05:01:47Z","abstract_excerpt":"The confluent hypergeometric functions (the Kummer functions) defined by ${}_{1}F_{1}(\\alpha;\\gamma;z):=\\sum_{n=0}^{\\infty}\\frac{(\\alpha)_{n}}{n!(\\gamma)_{n}}z^{n}\\ (\\gamma\\neq 0,-1,-2,\\cdots)$, which are of many properties and great applications in statistics, mathematical physics, engineering and so on, have been given. In this paper, we investigate some new properties of ${}_{1}F_{1}(\\alpha;\\gamma;z)$ from the perspective of value distribution theory. Specifically, two different growth orders are obtained for $\\alpha\\in \\mathbb{Z}_{\\leq 0}$ and $\\alpha\\not\\in \\mathbb{Z}_{\\leq 0}$, which are"},"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":"1509.06465","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-09-22T05:01:47Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"deaa622fb822c9f630c53d8090f06d7e9ef5fbc59fd257fecd01e826d7d40c92","abstract_canon_sha256":"3cf6ee7c676396bfd0978c92f7125df9014ee715aff072f4df1f3b4c8b2cbc6f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:20.203034Z","signature_b64":"dd4/HdgSgWX2AnVJjELR3xIEtKvsL4UzglriIwGAj8FrmmTmPVXLpaeOZe882oQ9kOlnFmw9bYlyTK4x+hxbAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cf1174ab761dd983adc82f5905bce6d42ea0ad534d1efd0b643b4c937d9a36d","last_reissued_at":"2026-05-18T01:32:20.202420Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:20.202420Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some new properties of Confluent Hypergeometric Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CV","authors_text":"Wei-Chuan Lin, Xu-Dan Luo","submitted_at":"2015-09-22T05:01:47Z","abstract_excerpt":"The confluent hypergeometric functions (the Kummer functions) defined by ${}_{1}F_{1}(\\alpha;\\gamma;z):=\\sum_{n=0}^{\\infty}\\frac{(\\alpha)_{n}}{n!(\\gamma)_{n}}z^{n}\\ (\\gamma\\neq 0,-1,-2,\\cdots)$, which are of many properties and great applications in statistics, mathematical physics, engineering and so on, have been given. In this paper, we investigate some new properties of ${}_{1}F_{1}(\\alpha;\\gamma;z)$ from the perspective of value distribution theory. Specifically, two different growth orders are obtained for $\\alpha\\in \\mathbb{Z}_{\\leq 0}$ and $\\alpha\\not\\in \\mathbb{Z}_{\\leq 0}$, which are"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06465","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":"1509.06465","created_at":"2026-05-18T01:32:20.202510+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.06465v1","created_at":"2026-05-18T01:32:20.202510+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.06465","created_at":"2026-05-18T01:32:20.202510+00:00"},{"alias_kind":"pith_short_12","alias_value":"TTYROSVXMHOZ","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"TTYROSVXMHOZQOW4","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"TTYROSVX","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/TTYROSVXMHOZQOW4QL2ZAW6ONV","json":"https://pith.science/pith/TTYROSVXMHOZQOW4QL2ZAW6ONV.json","graph_json":"https://pith.science/api/pith-number/TTYROSVXMHOZQOW4QL2ZAW6ONV/graph.json","events_json":"https://pith.science/api/pith-number/TTYROSVXMHOZQOW4QL2ZAW6ONV/events.json","paper":"https://pith.science/paper/TTYROSVX"},"agent_actions":{"view_html":"https://pith.science/pith/TTYROSVXMHOZQOW4QL2ZAW6ONV","download_json":"https://pith.science/pith/TTYROSVXMHOZQOW4QL2ZAW6ONV.json","view_paper":"https://pith.science/paper/TTYROSVX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.06465&json=true","fetch_graph":"https://pith.science/api/pith-number/TTYROSVXMHOZQOW4QL2ZAW6ONV/graph.json","fetch_events":"https://pith.science/api/pith-number/TTYROSVXMHOZQOW4QL2ZAW6ONV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TTYROSVXMHOZQOW4QL2ZAW6ONV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TTYROSVXMHOZQOW4QL2ZAW6ONV/action/storage_attestation","attest_author":"https://pith.science/pith/TTYROSVXMHOZQOW4QL2ZAW6ONV/action/author_attestation","sign_citation":"https://pith.science/pith/TTYROSVXMHOZQOW4QL2ZAW6ONV/action/citation_signature","submit_replication":"https://pith.science/pith/TTYROSVXMHOZQOW4QL2ZAW6ONV/action/replication_record"}},"created_at":"2026-05-18T01:32:20.202510+00:00","updated_at":"2026-05-18T01:32:20.202510+00:00"}