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Shorey","submitted_at":"2013-06-04T12:32:22Z","abstract_excerpt":"For a rational $q=u+\\frac{\\alpha}{d}$ with $u, \\alpha, d\\in \\ACOBZ$ with $u\\ge 0, 1\\le \\alpha<d$, $\\gcd(\\alpha, d)=1$, the \\emph{generalized Hermite-Laguerre polynomials $G_q(x)$} are defined by \\begin{align*} G_q(x)&=a_nx^n+a_{n-1}(\\alpha +(n-1+u)d)x^{n-1}+\\cdots\\\\ &\\quad+a_1\\left(\\prod^{n-1}_{i=1}(\\alpha +(i+u)d)\\right)x+a_0 \\left(\\prod^{n-1}_{i=0}(\\alpha +(i+u)d)\\right) \\end{align*} where $a_0, a_1, \\cdots, a_n$ are arbitrary integers. We prove some irreducibility results of $G_q(x)$ when $q\\in \\{\\frac{1}{3}, \\frac{2}{3}\\}$ and extend some of the earlier irreducibility results when $q$ of t"},"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":"1306.0745","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-06-04T12:32:22Z","cross_cats_sorted":[],"title_canon_sha256":"63ec0345ec5969151266617e793c3e59f7ee3defb575553f1bc1d2c6d7f1e684","abstract_canon_sha256":"4193e916e97ca982b87fdfb5032bf971881c8d4735d1c8c3d890bed82b9cc018"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:53.312099Z","signature_b64":"7ax5Jrquf5iHefWJSxWb5SsT5DCzjCqPapXujqm1gq1jIXs0Ne9HDcPdZszwgjwVU7PLCjsca6x6vMJp/zQlBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eafc2f1fc7b2ff8f7e3fdbcfec569bfa917c23a9e1666fbd509a6af52927e086","last_reissued_at":"2026-05-18T03:21:53.311597Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:53.311597Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Irreducibility of generalized Hermite-Laguerre polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Shanta Laishram, T. N. Shorey","submitted_at":"2013-06-04T12:32:22Z","abstract_excerpt":"For a rational $q=u+\\frac{\\alpha}{d}$ with $u, \\alpha, d\\in \\ACOBZ$ with $u\\ge 0, 1\\le \\alpha<d$, $\\gcd(\\alpha, d)=1$, the \\emph{generalized Hermite-Laguerre polynomials $G_q(x)$} are defined by \\begin{align*} G_q(x)&=a_nx^n+a_{n-1}(\\alpha +(n-1+u)d)x^{n-1}+\\cdots\\\\ &\\quad+a_1\\left(\\prod^{n-1}_{i=1}(\\alpha +(i+u)d)\\right)x+a_0 \\left(\\prod^{n-1}_{i=0}(\\alpha +(i+u)d)\\right) \\end{align*} where $a_0, a_1, \\cdots, a_n$ are arbitrary integers. We prove some irreducibility results of $G_q(x)$ when $q\\in \\{\\frac{1}{3}, \\frac{2}{3}\\}$ and extend some of the earlier irreducibility results when $q$ of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0745","kind":"arxiv","version":1},"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":"1306.0745","created_at":"2026-05-18T03:21:53.311661+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.0745v1","created_at":"2026-05-18T03:21:53.311661+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0745","created_at":"2026-05-18T03:21:53.311661+00:00"},{"alias_kind":"pith_short_12","alias_value":"5L6C6H6HWL7Y","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_16","alias_value":"5L6C6H6HWL7Y67R7","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_8","alias_value":"5L6C6H6H","created_at":"2026-05-18T12:27:34.582898+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/5L6C6H6HWL7Y67R73PH6YVU37K","json":"https://pith.science/pith/5L6C6H6HWL7Y67R73PH6YVU37K.json","graph_json":"https://pith.science/api/pith-number/5L6C6H6HWL7Y67R73PH6YVU37K/graph.json","events_json":"https://pith.science/api/pith-number/5L6C6H6HWL7Y67R73PH6YVU37K/events.json","paper":"https://pith.science/paper/5L6C6H6H"},"agent_actions":{"view_html":"https://pith.science/pith/5L6C6H6HWL7Y67R73PH6YVU37K","download_json":"https://pith.science/pith/5L6C6H6HWL7Y67R73PH6YVU37K.json","view_paper":"https://pith.science/paper/5L6C6H6H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.0745&json=true","fetch_graph":"https://pith.science/api/pith-number/5L6C6H6HWL7Y67R73PH6YVU37K/graph.json","fetch_events":"https://pith.science/api/pith-number/5L6C6H6HWL7Y67R73PH6YVU37K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5L6C6H6HWL7Y67R73PH6YVU37K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5L6C6H6HWL7Y67R73PH6YVU37K/action/storage_attestation","attest_author":"https://pith.science/pith/5L6C6H6HWL7Y67R73PH6YVU37K/action/author_attestation","sign_citation":"https://pith.science/pith/5L6C6H6HWL7Y67R73PH6YVU37K/action/citation_signature","submit_replication":"https://pith.science/pith/5L6C6H6HWL7Y67R73PH6YVU37K/action/replication_record"}},"created_at":"2026-05-18T03:21:53.311661+00:00","updated_at":"2026-05-18T03:21:53.311661+00:00"}