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Guo","submitted_at":"2017-05-08T11:13:06Z","abstract_excerpt":"We prove that, for all positive integers $n_1, \\ldots, n_m$, $n_{m+1}=n_1$, and non-negative integers $j$ and $r$ with $j\\leqslant m$, the following two expressions \\begin{align*} &\\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\\brack n_1}^{-1}\\sum_{k=0}^{n_1} q^{j(k^2+k)-(2r+1)k}[2k+1]^{2r+1}\\prod_{i=1}^m {n_i+n_{i+1}+1\\brack n_i-k},\\\\[5pt] &\\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\\brack n_1}^{-1}\\sum_{k=0}^{n_1}(-1)^k q^{{k\\choose 2}+j(k^2+k)-2rk}[2k+1]^{2r+1}\\prod_{i=1}^m {n_i+n_{i+1}+1\\brack n_i-k} \\end{align*} are Laurent polynomials in $q$ with integer coefficients, where $[n]=1+q+\\cdots+q^{n-1}$ and ${n\\brack k"},"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":"1705.06236","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-08T11:13:06Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b5ec98f3255efa8f286b18cbed67e47560cd2faea7c86ec4fc0bf3cd82c07f17","abstract_canon_sha256":"d4acc74b252db908b73d95cab0449d14aaa00e871f49738c0256ec16d6fd6f0b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:18.365193Z","signature_b64":"a/exWu/KSzmgQmsCf8MsHiXfVwbvwJHcCu0hxxn14Gat65ivmFwGLZFqtkLgJO/tUCl3qAxpf8Bp67iU90FiAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e68ea0d69a9056574440551f7869c9bd0d48e005c6491ebc4b5691b2686881b9","last_reissued_at":"2026-05-18T00:44:18.364737Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:18.364737Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Factors of sums and alternating sums of products of $q$-binomial coefficients and powers of $q$-integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Su-Dan Wang, Victor J. 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Guo","submitted_at":"2017-05-08T11:13:06Z","abstract_excerpt":"We prove that, for all positive integers $n_1, \\ldots, n_m$, $n_{m+1}=n_1$, and non-negative integers $j$ and $r$ with $j\\leqslant m$, the following two expressions \\begin{align*} &\\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\\brack n_1}^{-1}\\sum_{k=0}^{n_1} q^{j(k^2+k)-(2r+1)k}[2k+1]^{2r+1}\\prod_{i=1}^m {n_i+n_{i+1}+1\\brack n_i-k},\\\\[5pt] &\\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\\brack n_1}^{-1}\\sum_{k=0}^{n_1}(-1)^k q^{{k\\choose 2}+j(k^2+k)-2rk}[2k+1]^{2r+1}\\prod_{i=1}^m {n_i+n_{i+1}+1\\brack n_i-k} \\end{align*} are Laurent polynomials in $q$ with integer coefficients, where $[n]=1+q+\\cdots+q^{n-1}$ and ${n\\brack k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.06236","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":"1705.06236","created_at":"2026-05-18T00:44:18.364809+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.06236v1","created_at":"2026-05-18T00:44:18.364809+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.06236","created_at":"2026-05-18T00:44:18.364809+00:00"},{"alias_kind":"pith_short_12","alias_value":"42HKBVU2SBLF","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"42HKBVU2SBLFORCA","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"42HKBVU2","created_at":"2026-05-18T12:30:58.224056+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/42HKBVU2SBLFORCAKUPXQ2OJXU","json":"https://pith.science/pith/42HKBVU2SBLFORCAKUPXQ2OJXU.json","graph_json":"https://pith.science/api/pith-number/42HKBVU2SBLFORCAKUPXQ2OJXU/graph.json","events_json":"https://pith.science/api/pith-number/42HKBVU2SBLFORCAKUPXQ2OJXU/events.json","paper":"https://pith.science/paper/42HKBVU2"},"agent_actions":{"view_html":"https://pith.science/pith/42HKBVU2SBLFORCAKUPXQ2OJXU","download_json":"https://pith.science/pith/42HKBVU2SBLFORCAKUPXQ2OJXU.json","view_paper":"https://pith.science/paper/42HKBVU2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.06236&json=true","fetch_graph":"https://pith.science/api/pith-number/42HKBVU2SBLFORCAKUPXQ2OJXU/graph.json","fetch_events":"https://pith.science/api/pith-number/42HKBVU2SBLFORCAKUPXQ2OJXU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/42HKBVU2SBLFORCAKUPXQ2OJXU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/42HKBVU2SBLFORCAKUPXQ2OJXU/action/storage_attestation","attest_author":"https://pith.science/pith/42HKBVU2SBLFORCAKUPXQ2OJXU/action/author_attestation","sign_citation":"https://pith.science/pith/42HKBVU2SBLFORCAKUPXQ2OJXU/action/citation_signature","submit_replication":"https://pith.science/pith/42HKBVU2SBLFORCAKUPXQ2OJXU/action/replication_record"}},"created_at":"2026-05-18T00:44:18.364809+00:00","updated_at":"2026-05-18T00:44:18.364809+00:00"}