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The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pairs are there that satisfy those relations? It turns out that the answer is 5 in general. We give the generic number of Leonard pairs for each Askey-Wilson type of Askey-Wilson relations."},"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":"math/0511509","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.QA","submitted_at":"2005-11-21T05:10:14Z","cross_cats_sorted":[],"title_canon_sha256":"8455dd49833c71f18c472b0c71ded93b04625a946fda473b3aa4e2f5a634a517","abstract_canon_sha256":"813097a22d9f3bdbf201eddca2ff6da3e3061f3d9d12525c4d7fa640764cf795"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:24.460485Z","signature_b64":"USkg0lFbblduEoGQ4HJalUNQw0CVKQTfdFIuYgHcvNntGBCqsgwv+QQphThsyxQXQfxT7I77wVAWNafTsq/cBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"afa3ed5d6e9a40b7e4ba142e64d765f64d2faa262f0b784b2c3546ff61f2aa97","last_reissued_at":"2026-05-18T03:11:24.459663Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:24.459663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Askey-Wilson relations and Leonard pairs","license":"","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Raimundas Vidunas","submitted_at":"2005-11-21T05:10:14Z","abstract_excerpt":"It is known that if $(A,B)$ is a Leonard pair, then the linear transformations $A$, $B$ satisfy the Askey-Wilson relations A^2 B - b A B A + B A^2 - g (A B+B A) - r B = h A^2 + w A + e I,\n  B^2 A - b B A B + A B^2 - h (A B+B A) - s A = g B^2 + w B + f I, for some scalars $b,g,h,r,s,w,e,f$. The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pairs are there that satisfy those relations? It turns out that the answer is 5 in general. We give the generic number of Leonard pairs for each Askey-Wilson type of Askey-Wilson relations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0511509","kind":"arxiv","version":2},"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":"math/0511509","created_at":"2026-05-18T03:11:24.459790+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0511509v2","created_at":"2026-05-18T03:11:24.459790+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0511509","created_at":"2026-05-18T03:11:24.459790+00:00"},{"alias_kind":"pith_short_12","alias_value":"V6R62XLOTJAL","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"V6R62XLOTJALPZF2","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"V6R62XLO","created_at":"2026-05-18T12:25:53.939244+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/V6R62XLOTJALPZF2CQXGJV3F6Z","json":"https://pith.science/pith/V6R62XLOTJALPZF2CQXGJV3F6Z.json","graph_json":"https://pith.science/api/pith-number/V6R62XLOTJALPZF2CQXGJV3F6Z/graph.json","events_json":"https://pith.science/api/pith-number/V6R62XLOTJALPZF2CQXGJV3F6Z/events.json","paper":"https://pith.science/paper/V6R62XLO"},"agent_actions":{"view_html":"https://pith.science/pith/V6R62XLOTJALPZF2CQXGJV3F6Z","download_json":"https://pith.science/pith/V6R62XLOTJALPZF2CQXGJV3F6Z.json","view_paper":"https://pith.science/paper/V6R62XLO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0511509&json=true","fetch_graph":"https://pith.science/api/pith-number/V6R62XLOTJALPZF2CQXGJV3F6Z/graph.json","fetch_events":"https://pith.science/api/pith-number/V6R62XLOTJALPZF2CQXGJV3F6Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V6R62XLOTJALPZF2CQXGJV3F6Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V6R62XLOTJALPZF2CQXGJV3F6Z/action/storage_attestation","attest_author":"https://pith.science/pith/V6R62XLOTJALPZF2CQXGJV3F6Z/action/author_attestation","sign_citation":"https://pith.science/pith/V6R62XLOTJALPZF2CQXGJV3F6Z/action/citation_signature","submit_replication":"https://pith.science/pith/V6R62XLOTJALPZF2CQXGJV3F6Z/action/replication_record"}},"created_at":"2026-05-18T03:11:24.459790+00:00","updated_at":"2026-05-18T03:11:24.459790+00:00"}