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The scalar sequence is unique if the dimension of $V$ is at least 4. If $c,c*,t,t*$ are scalars and $t,t*$ are not zero, then $(tA+c,t*B+c*)$ is a Leonard pair on $V$ as well. These affine transformations can be used to bring the Leonard pair or its Askey-Wilson relations into 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":"math/0505041","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.RA","submitted_at":"2005-05-03T03:24:40Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"dc63d1df3b822712f4b275d497e4703bf1a611acdc1184bdfb570193f95eae5d","abstract_canon_sha256":"90abcdb98b198942e2be8aaaa8d9cd7ad4ff3e42ff8d88dc49436a3e5b874357"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:24.470633Z","signature_b64":"BCMkI1cqPbq+hbeH6XCGUZMTQ8CwO1N7yRUf6kR/w67UYBSJe7tw7/ne/izfqsQWOm4Cvu4U0c9rCACrbA54Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b497954e80996ccf921eb7dd38aa9f96d439f50fd4ee46e4da3d42ed0d8bc6b","last_reissued_at":"2026-05-18T03:11:24.469869Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:24.469869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normalized Leonard pairs and Askey-Wilson relations","license":"","headline":"","cross_cats":["math.QA"],"primary_cat":"math.RA","authors_text":"Raimundas Vidunas","submitted_at":"2005-05-03T03:24:40Z","abstract_excerpt":"Let $V$ denote a vector space with finite positive dimension, and let $(A,B)$ denote a Leonard pair on $V$. As is known, the linear transformations $A,B$ satisfy the Askey-Wilson relations A^2B -bABA +BA^2 -g(AB+BA) -rB = hA^2 +wA +eI,\n  B^2A -bBAB +AB^2 -h(AB+BA) -sA = gB^2 +wB +fI, for some scalars $b,g,h,r,s,w,e,f$. The scalar sequence is unique if the dimension of $V$ is at least 4. If $c,c*,t,t*$ are scalars and $t,t*$ are not zero, then $(tA+c,t*B+c*)$ is a Leonard pair on $V$ as well. 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