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We show that the columns of the 2-D Riordan array are eventually periodic sequences, where a circulant matrix generated by the coefficients of $p_3(t)$ determines the behavior of this periodicity as the column index grows indefinitely. Furthermore, we prove that the preperiodic column partial sums of the 2-D array are periodic, and present a family o"},"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":"2607.13442","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-15T04:58:49Z","cross_cats_sorted":[],"title_canon_sha256":"acee504e8485c76944d72bebcbdc5e5cd715f91dd129d221104b2073d13c8595","abstract_canon_sha256":"fb45a732831a90fe11d885621a5e1694191587534f1322a392942fb8acc32c02"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-16T01:22:12.944491Z","signature_b64":"xbVZkhwktqKISRkJHl7+lnWnquT4W7EBJrhIkdx3BvBpKZhky7nSC2ueT9kcV1QfwPoF0icjF4+5ZPLMI4GtBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"79bc82dcc73b1c342d727dedaff69f70f7ae19ae8b7e193c8fe78ff359b8fa57","last_reissued_at":"2026-07-16T01:22:12.943531Z","signature_status":"signed_v1","first_computed_at":"2026-07-16T01:22:12.943531Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Periodicities in the Riordan arrays of polynomials over finite fields","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Derek E. Bellamy, Eva N. Pflomm, Nikolai A. Krylov","submitted_at":"2026-07-15T04:58:49Z","abstract_excerpt":"We study periodicity properties of the 2-D $\\bigl(p_1(t)/p_2(t),\\, tp_3(t)\\bigr)$ and 3-D $\\bigl(p_1(t)/p_2(t),\\, tp_3(t),\\, p_4(t)\\bigr)$ Riordan arrays over a finite field ${\\mathbb F}_q$, where each $p_i(t)$ is a polynomial with $p_i(0)\\neq 0$. We show that the columns of the 2-D Riordan array are eventually periodic sequences, where a circulant matrix generated by the coefficients of $p_3(t)$ determines the behavior of this periodicity as the column index grows indefinitely. 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