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Our first algorithm searches for solutions of the form \\[ \\exp(\\int r \\, dx)\\cdot{_{2}F_1}(a_1,a_2;b_1;f) \\] where $r,f \\in \\overline{\\mathbb{Q}(x)}$, and $a_1,a_2,b_1 \\in \\mathbb{Q}$. It uses modular reduction and Hensel lifting. Our second algorithm tries to find solutions in the form \\[ \\exp(\\int r \\, dx)\\cdot \\left( r_0 \\cdot{_{2}F_1}(a_1,a_2;b_1;f) + r_1 \\cdot{_{2}F_1}'(a_1,a_2;b_1;f) \\right) \\] where $r_0, r_1 \\in \\overline{\\mathbb{Q}(x)}$, 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":"1606.01576","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SC","submitted_at":"2016-06-05T22:41:58Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"5d4d00a3f6d6203b9f190ff42a0853afbce23f30157b22d431c82102739f5e77","abstract_canon_sha256":"37b8d70606238faa7670e1b0229fd3af8576130354e7b483565c0fe54884f02b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:53.947346Z","signature_b64":"G+pVMABgMAZ5jpSTyLJSr0WHR217a5BQco4PpTKLmv3iHO9ReYHCIIpy6eIscdzg1JTEqYQYcF5KrUCqfe9OCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2d3608c5bdb0831df208acb08c6eb59fdc3ee47fa27a167668820cdc6e89c1c","last_reissued_at":"2026-05-18T01:12:53.946994Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:53.946994Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions and Integral Bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"cs.SC","authors_text":"Erdal Imamoglu, Mark van Hoeij","submitted_at":"2016-06-05T22:41:58Z","abstract_excerpt":"We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form \\[ \\exp(\\int r \\, dx)\\cdot{_{2}F_1}(a_1,a_2;b_1;f) \\] where $r,f \\in \\overline{\\mathbb{Q}(x)}$, and $a_1,a_2,b_1 \\in \\mathbb{Q}$. It uses modular reduction and Hensel lifting. Our second algorithm tries to find solutions in the form \\[ \\exp(\\int r \\, dx)\\cdot \\left( r_0 \\cdot{_{2}F_1}(a_1,a_2;b_1;f) + r_1 \\cdot{_{2}F_1}'(a_1,a_2;b_1;f) \\right) \\] where $r_0, r_1 \\in \\overline{\\mathbb{Q}(x)}$, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.01576","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":"1606.01576","created_at":"2026-05-18T01:12:53.947050+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.01576v1","created_at":"2026-05-18T01:12:53.947050+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.01576","created_at":"2026-05-18T01:12:53.947050+00:00"},{"alias_kind":"pith_short_12","alias_value":"ULJWBDC33MED","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"ULJWBDC33MEDDXZA","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"ULJWBDC3","created_at":"2026-05-18T12:30:46.583412+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/ULJWBDC33MEDDXZARLFQRRXLLH","json":"https://pith.science/pith/ULJWBDC33MEDDXZARLFQRRXLLH.json","graph_json":"https://pith.science/api/pith-number/ULJWBDC33MEDDXZARLFQRRXLLH/graph.json","events_json":"https://pith.science/api/pith-number/ULJWBDC33MEDDXZARLFQRRXLLH/events.json","paper":"https://pith.science/paper/ULJWBDC3"},"agent_actions":{"view_html":"https://pith.science/pith/ULJWBDC33MEDDXZARLFQRRXLLH","download_json":"https://pith.science/pith/ULJWBDC33MEDDXZARLFQRRXLLH.json","view_paper":"https://pith.science/paper/ULJWBDC3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.01576&json=true","fetch_graph":"https://pith.science/api/pith-number/ULJWBDC33MEDDXZARLFQRRXLLH/graph.json","fetch_events":"https://pith.science/api/pith-number/ULJWBDC33MEDDXZARLFQRRXLLH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ULJWBDC33MEDDXZARLFQRRXLLH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ULJWBDC33MEDDXZARLFQRRXLLH/action/storage_attestation","attest_author":"https://pith.science/pith/ULJWBDC33MEDDXZARLFQRRXLLH/action/author_attestation","sign_citation":"https://pith.science/pith/ULJWBDC33MEDDXZARLFQRRXLLH/action/citation_signature","submit_replication":"https://pith.science/pith/ULJWBDC33MEDDXZARLFQRRXLLH/action/replication_record"}},"created_at":"2026-05-18T01:12:53.947050+00:00","updated_at":"2026-05-18T01:12:53.947050+00:00"}