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We consider an ordered pair of linear transformations $A: V\\to V$ and $A^*: V\\to V$ that satisfy (i) and (ii) below:\n  (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal.\n  (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal.\n  We call such a pair a Leonard pair on $V$. In this paper, we characterize the Leonard pairs using th"},"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":"0911.0098","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2009-10-31T18:00:33Z","cross_cats_sorted":[],"title_canon_sha256":"77106ccaf159b92a265094e3ecae268f52afa93d46be95b4f7a04e5bb59b188a","abstract_canon_sha256":"d04cc0763fb15b4f09df6491465c1117ae9023b834799d776e275f959de038f8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T16:21:35.250266Z","signature_b64":"qsq3pXuQfgfaoAz1A9/HB8a+DHrZoOC1XkEaOI9BZ7GIZPPALg9KqyYDSzEWyelB00HZ+MomQr3hRyVhXWQyDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab2e51d873d736e68a5bec93f63e7289197fbc8c8b45aa659b94cb81b61bd136","last_reissued_at":"2026-07-04T16:21:35.249874Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T16:21:35.249874Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A characterization of Leonard pairs using the notion of a tail","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Edward Hanson","submitted_at":"2009-10-31T18:00:33Z","abstract_excerpt":"Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\\to V$ and $A^*: V\\to V$ that satisfy (i) and (ii) below:\n  (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal.\n  (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal.\n  We call such a pair a Leonard pair on $V$. 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