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Guo","submitted_at":"2009-10-19T14:23:19Z","abstract_excerpt":"Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$ \\sum_{k=0}^{n-1}(-1)^kq^{-{k+1\\choose 2}}{2k\\brack k}_q \\equiv (\\frac{n}{5}) q^{-\\lfloor n^4/5\\rfloor} \\pmod{\\Phi_n(q)}, $$ where $\\big(\\frac{n}{p}\\big)$ is the Legendre symbol and $\\Phi_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that $$ \\sum_{k=0}^{3^a m-1} q^{k}{2k\\brack k}_q &\\equiv 0 \\pmod{(1-q^{3^a})/(1-q)}, \\sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\\choose 2}}{2k\\brack k}_q &\\equiv 0 \\pmod{(1-q^{5^a})/(1-q)}, $$ for $"},"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":"0910.3563","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-10-19T14:23:19Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6740252328526f843b98d908d83cd0c453b7fae7242af54ca710790f4c094393","abstract_canon_sha256":"a5a8ad1283a95bef637b5ddf378994e97e75078d589e191a09d06a047a38a7ac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:01.530802Z","signature_b64":"8dv3iPrkDhKqRIp6ECyY2skeCJU5q+5ZSuMwoOAN4IIjJxnk19fivEUK6+EzkRJ1avKxnPxIgmYAbZW84FGSAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88ec6e70971fb886741368768bf9b7966c08e4ed8075029efbb73e941f33516d","last_reissued_at":"2026-05-18T04:26:01.530362Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:01.530362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some congruences involving central q-binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Jiang Zeng, Victor J. W. Guo","submitted_at":"2009-10-19T14:23:19Z","abstract_excerpt":"Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$ \\sum_{k=0}^{n-1}(-1)^kq^{-{k+1\\choose 2}}{2k\\brack k}_q \\equiv (\\frac{n}{5}) q^{-\\lfloor n^4/5\\rfloor} \\pmod{\\Phi_n(q)}, $$ where $\\big(\\frac{n}{p}\\big)$ is the Legendre symbol and $\\Phi_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that $$ \\sum_{k=0}^{3^a m-1} q^{k}{2k\\brack k}_q &\\equiv 0 \\pmod{(1-q^{3^a})/(1-q)}, \\sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\\choose 2}}{2k\\brack k}_q &\\equiv 0 \\pmod{(1-q^{5^a})/(1-q)}, $$ for $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.3563","kind":"arxiv","version":3},"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":"0910.3563","created_at":"2026-05-18T04:26:01.530431+00:00"},{"alias_kind":"arxiv_version","alias_value":"0910.3563v3","created_at":"2026-05-18T04:26:01.530431+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.3563","created_at":"2026-05-18T04:26:01.530431+00:00"},{"alias_kind":"pith_short_12","alias_value":"RDWG44EXD64I","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_16","alias_value":"RDWG44EXD64IM5AT","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_8","alias_value":"RDWG44EX","created_at":"2026-05-18T12:26:01.383474+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/RDWG44EXD64IM5ATNB3IX6NXSZ","json":"https://pith.science/pith/RDWG44EXD64IM5ATNB3IX6NXSZ.json","graph_json":"https://pith.science/api/pith-number/RDWG44EXD64IM5ATNB3IX6NXSZ/graph.json","events_json":"https://pith.science/api/pith-number/RDWG44EXD64IM5ATNB3IX6NXSZ/events.json","paper":"https://pith.science/paper/RDWG44EX"},"agent_actions":{"view_html":"https://pith.science/pith/RDWG44EXD64IM5ATNB3IX6NXSZ","download_json":"https://pith.science/pith/RDWG44EXD64IM5ATNB3IX6NXSZ.json","view_paper":"https://pith.science/paper/RDWG44EX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0910.3563&json=true","fetch_graph":"https://pith.science/api/pith-number/RDWG44EXD64IM5ATNB3IX6NXSZ/graph.json","fetch_events":"https://pith.science/api/pith-number/RDWG44EXD64IM5ATNB3IX6NXSZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RDWG44EXD64IM5ATNB3IX6NXSZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RDWG44EXD64IM5ATNB3IX6NXSZ/action/storage_attestation","attest_author":"https://pith.science/pith/RDWG44EXD64IM5ATNB3IX6NXSZ/action/author_attestation","sign_citation":"https://pith.science/pith/RDWG44EXD64IM5ATNB3IX6NXSZ/action/citation_signature","submit_replication":"https://pith.science/pith/RDWG44EXD64IM5ATNB3IX6NXSZ/action/replication_record"}},"created_at":"2026-05-18T04:26:01.530431+00:00","updated_at":"2026-05-18T04:26:01.530431+00:00"}