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From this formulation we construct a pair of sequences, $\\{P_n(e_1)\\}$ and $\\{Q_m(e_2)\\}$, of bi-orthogonal polynomials (BiOPS), that is, they satisfy $\\mathcal{L}(P_n(e_1)\\otimes Q_m(e_2))=\\Lambda_n\\delta_{n,m}$. The existence of the sequences arises from the determinant of a Pascal triangle like matrix of polynomials. The polynomia"},"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":"1412.7235","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-12-23T02:06:13Z","cross_cats_sorted":["math.MP","math.RA","math.RT"],"title_canon_sha256":"12f9e76bac2a3f99f75d50f63a23fe7ea3439b155b6625430378adf284f0cf0d","abstract_canon_sha256":"6c2e7900a4f3bcf66c6779211920be86b91f382b26f2fec3c3e5be14d64b7c85"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:01.661995Z","signature_b64":"CmsYBS0LQw8pnvI5qA2OFcSvbVUBmY8G/cozxNwDvq9Hlv8XDXyl9rySuN1lqXirKVEEN/JHRi9eMfhVuza8DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7deccdeed84ce813154c894868a2329186861322226c3ed66d841f7eef1861bd","last_reissued_at":"2026-05-18T00:14:01.661271Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:01.661271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bi-orthogonal Polynomial Sequences and the Asymmetric Simple Exclusion Process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.RA","math.RT"],"primary_cat":"math-ph","authors_text":"Richard Brak, William Moore","submitted_at":"2014-12-23T02:06:13Z","abstract_excerpt":"We reformulate the Corteel-Williams equations for the stationary state of the two parameter Asymmetric Simple Exclusion Process (TASEP) as a linear map $\\mathcal{L}(\\,\\cdot\\,)$, acting on a tensor algebra built from a rank two free module with basis $\\{e_1,e_2\\}$. 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