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We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\\leq j\\leq m-1, {n_1+n_{m}\\brack n_1}^{-1}\\sum_{k=-n_1}^{n_1}(-1)^kq^{jk^2+{k\\choose 2}} \\prod_{i=1}^m {n_i+n_{i+1}\\brack n_i+k}\\in \\N[q], which generalizes a result of Calkin [Acta Arith. 86 (1998), 17--26]. Moreover, we show that for all positive integers n, r and j, {2n\\brack n}^{-1}{2j\\brack j} \\sum_{k=j}^n(-1)^{n-k}q^{A}\\frac{1-q^{2k+1}}{1-q^{n+k+1}} {2n\\brack n-k}{k+j\\brack k-j}^r\\in N[q], where A=(r-1){n\\c"},"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":"math/0511635","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2005-11-25T19:47:22Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"8a7007cb60173b8d74d521cd92ada97d94269a8f4af8d12dc51af14eb5314db6","abstract_canon_sha256":"261bea666bcdc57e6f6b30e5fd8b9370ca2becf7dda8ddaede8cb2b2828ece0a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:24.625157Z","signature_b64":"oiZ9/GkeBP3XSFWQLgc32jWuCR8KnDKYailY8Fst6U/upAPNjAFrwjISTrGbLEpoDhEWqVn+jYj8MlmGmzfkDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88827f50d844415f6554b2644561917d2743051777afd0ac7e578374597d2a76","last_reissued_at":"2026-05-18T01:38:24.624474Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:24.624474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Factors of Alternating Sums of Products of Binomial and q-Binomial Coefficients","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Frederic Jouhet, Jiang Zeng, Victor J. W. Guo","submitted_at":"2005-11-25T19:47:22Z","abstract_excerpt":"In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\\leq j\\leq m-1, {n_1+n_{m}\\brack n_1}^{-1}\\sum_{k=-n_1}^{n_1}(-1)^kq^{jk^2+{k\\choose 2}} \\prod_{i=1}^m {n_i+n_{i+1}\\brack n_i+k}\\in \\N[q], which generalizes a result of Calkin [Acta Arith. 86 (1998), 17--26]. Moreover, we show that for all positive integers n, r and j, {2n\\brack n}^{-1}{2j\\brack j} \\sum_{k=j}^n(-1)^{n-k}q^{A}\\frac{1-q^{2k+1}}{1-q^{n+k+1}} {2n\\brack n-k}{k+j\\brack k-j}^r\\in N[q], where A=(r-1){n\\c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0511635","kind":"arxiv","version":4},"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":"math/0511635","created_at":"2026-05-18T01:38:24.624579+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0511635v4","created_at":"2026-05-18T01:38:24.624579+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0511635","created_at":"2026-05-18T01:38:24.624579+00:00"},{"alias_kind":"pith_short_12","alias_value":"RCBH6UGYIRAV","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_16","alias_value":"RCBH6UGYIRAV6ZKU","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_8","alias_value":"RCBH6UGY","created_at":"2026-05-18T12:25:53.335082+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/RCBH6UGYIRAV6ZKUWJSEKYMRPU","json":"https://pith.science/pith/RCBH6UGYIRAV6ZKUWJSEKYMRPU.json","graph_json":"https://pith.science/api/pith-number/RCBH6UGYIRAV6ZKUWJSEKYMRPU/graph.json","events_json":"https://pith.science/api/pith-number/RCBH6UGYIRAV6ZKUWJSEKYMRPU/events.json","paper":"https://pith.science/paper/RCBH6UGY"},"agent_actions":{"view_html":"https://pith.science/pith/RCBH6UGYIRAV6ZKUWJSEKYMRPU","download_json":"https://pith.science/pith/RCBH6UGYIRAV6ZKUWJSEKYMRPU.json","view_paper":"https://pith.science/paper/RCBH6UGY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0511635&json=true","fetch_graph":"https://pith.science/api/pith-number/RCBH6UGYIRAV6ZKUWJSEKYMRPU/graph.json","fetch_events":"https://pith.science/api/pith-number/RCBH6UGYIRAV6ZKUWJSEKYMRPU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RCBH6UGYIRAV6ZKUWJSEKYMRPU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RCBH6UGYIRAV6ZKUWJSEKYMRPU/action/storage_attestation","attest_author":"https://pith.science/pith/RCBH6UGYIRAV6ZKUWJSEKYMRPU/action/author_attestation","sign_citation":"https://pith.science/pith/RCBH6UGYIRAV6ZKUWJSEKYMRPU/action/citation_signature","submit_replication":"https://pith.science/pith/RCBH6UGYIRAV6ZKUWJSEKYMRPU/action/replication_record"}},"created_at":"2026-05-18T01:38:24.624579+00:00","updated_at":"2026-05-18T01:38:24.624579+00:00"}