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Via discrete Fourier transforms, we establish the following two identities: $$\\det\\left[\\left[\\left\\lfloor\\frac{aj-(a+1)k}n\\right\\rfloor\\right]_q\\right]_{1\\leqslant j,k\\leqslant n}=-\\left(\\frac{a(a+1)}n\\right)q^{(1-3n)/2}$$ and $$\\det\\left[\\left[\\left\\lceil\\frac{(a+1)j-ak}n\\right\\rceil\\right]_q\\right]_{1\\leqslant j,k\\leqslant n}=\\left(\\frac{a(a+1)}n\\right)q^{(n-1)/2},$$ where $(\\frac{\\cdot}n)$ denotes the Jacobi symbol."},"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":"2605.16240","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-15T17:48:24Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"77bbdb5dcbb564500479e646e2fa253e32be4a8170c0013cc56db8a7a90eade3","abstract_canon_sha256":"48fdc696dd9abddc4660923951ef49946974a9c30c480b3a424e75f7c2a61e6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:05:48.088214Z","signature_b64":"+4XzIENeW7fBsyLlAjroKGT+569wPcuwcihikIVfewkbloeoBvuxanCfGJMsB9SXuuRfSx1vY0v1buiF6JeWBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95bd9cddd5f6402edce76d665ac06d737070f621b7e9d2fe7ff5226ed8bccc06","last_reissued_at":"2026-05-20T00:05:48.087543Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:05:48.087543Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Evaluation of two determinants involving $q$-integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Zhi-Wei Sun","submitted_at":"2026-05-15T17:48:24Z","abstract_excerpt":"The $q$-analogue of an integer $m$ is given by $[m]_q=(1-q^m)/(1-q)$. Let $a$ be an integer, and let $n$ be a positive odd integer. Via discrete Fourier transforms, we establish the following two identities: $$\\det\\left[\\left[\\left\\lfloor\\frac{aj-(a+1)k}n\\right\\rfloor\\right]_q\\right]_{1\\leqslant j,k\\leqslant n}=-\\left(\\frac{a(a+1)}n\\right)q^{(1-3n)/2}$$ and $$\\det\\left[\\left[\\left\\lceil\\frac{(a+1)j-ak}n\\right\\rceil\\right]_q\\right]_{1\\leqslant j,k\\leqslant n}=\\left(\\frac{a(a+1)}n\\right)q^{(n-1)/2},$$ where $(\\frac{\\cdot}n)$ denotes the Jacobi symbol."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.16240","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16240/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2605.16240","created_at":"2026-05-20T00:05:48.087652+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.16240v2","created_at":"2026-05-20T00:05:48.087652+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.16240","created_at":"2026-05-20T00:05:48.087652+00:00"},{"alias_kind":"pith_short_12","alias_value":"SW6ZZXOV6ZAC","created_at":"2026-05-20T00:05:48.087652+00:00"},{"alias_kind":"pith_short_16","alias_value":"SW6ZZXOV6ZAC5XHH","created_at":"2026-05-20T00:05:48.087652+00:00"},{"alias_kind":"pith_short_8","alias_value":"SW6ZZXOV","created_at":"2026-05-20T00:05:48.087652+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/SW6ZZXOV6ZAC5XHHNVTFVQDNON","json":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON.json","graph_json":"https://pith.science/api/pith-number/SW6ZZXOV6ZAC5XHHNVTFVQDNON/graph.json","events_json":"https://pith.science/api/pith-number/SW6ZZXOV6ZAC5XHHNVTFVQDNON/events.json","paper":"https://pith.science/paper/SW6ZZXOV"},"agent_actions":{"view_html":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON","download_json":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON.json","view_paper":"https://pith.science/paper/SW6ZZXOV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.16240&json=true","fetch_graph":"https://pith.science/api/pith-number/SW6ZZXOV6ZAC5XHHNVTFVQDNON/graph.json","fetch_events":"https://pith.science/api/pith-number/SW6ZZXOV6ZAC5XHHNVTFVQDNON/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON/action/storage_attestation","attest_author":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON/action/author_attestation","sign_citation":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON/action/citation_signature","submit_replication":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON/action/replication_record"}},"created_at":"2026-05-20T00:05:48.087652+00:00","updated_at":"2026-05-20T00:05:48.087652+00:00"}