{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:M7IGVK6PJFC5LDXPCKV2ECTCTR","short_pith_number":"pith:M7IGVK6P","schema_version":"1.0","canonical_sha256":"67d06aabcf4945d58eef12aba20a629c6d42dcae7b3660302a5fff17dc712681","source":{"kind":"arxiv","id":"1703.00003","version":1},"attestation_state":"computed","paper":{"title":"Factors of alternating sums of powers of $q$-Narayana numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Qiang-Qiang Jiang, Victor J. W. Guo","submitted_at":"2017-02-28T01:50:31Z","abstract_excerpt":"The $q$-Narayana numbers $N_q(n,k)$ and $q$-Catalan numbers $C_n(q)$ are respectively defined by $$ N_q(n,k)=\\frac{1-q}{1-q^n}{n\\brack k}{n\\brack k-1}\\quad\\text{and}\\quad C_n(q)=\\frac{1-q}{1-q^{n+1}}{2n\\brack n}, $$ where ${n\\brack k}=\\prod_{i=1}^{k}\\frac{1-q^{n-i+1}}{1-q^i}$. We prove that, for any positive integers $n$ and $r$, there holds \\begin{align*} \\sum_{k=-n}^{n}(-1)^{k}q^{jk^2+{k\\choose 2}}N_q(2n+1,n+k+1)^r \\equiv 0 \\pmod{C_n(q)}, \\end{align*} where $0\\leqslant j\\leqslant 2r-1$. We also propose several related conjectures."},"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":"1703.00003","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-02-28T01:50:31Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6f2ab70e45b7fbbaffe66c71852a6d0caeba11cbfb5980d250a145f7b30f730b","abstract_canon_sha256":"5c2bc9a67179d84dd691eccaad46662b0d7a8a9cb37a26c49330d535fafb3b49"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:46.255182Z","signature_b64":"6x0KVdeNkL516GAnCkitTxKB8tugJDGXZsGg3Dl0rbSoS9eTowa95YzmON839DAZH8ejO+c2gahV5aiNXHaDDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67d06aabcf4945d58eef12aba20a629c6d42dcae7b3660302a5fff17dc712681","last_reissued_at":"2026-05-18T00:49:46.253952Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:46.253952Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Factors of alternating sums of powers of $q$-Narayana numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Qiang-Qiang Jiang, Victor J. W. Guo","submitted_at":"2017-02-28T01:50:31Z","abstract_excerpt":"The $q$-Narayana numbers $N_q(n,k)$ and $q$-Catalan numbers $C_n(q)$ are respectively defined by $$ N_q(n,k)=\\frac{1-q}{1-q^n}{n\\brack k}{n\\brack k-1}\\quad\\text{and}\\quad C_n(q)=\\frac{1-q}{1-q^{n+1}}{2n\\brack n}, $$ where ${n\\brack k}=\\prod_{i=1}^{k}\\frac{1-q^{n-i+1}}{1-q^i}$. We prove that, for any positive integers $n$ and $r$, there holds \\begin{align*} \\sum_{k=-n}^{n}(-1)^{k}q^{jk^2+{k\\choose 2}}N_q(2n+1,n+k+1)^r \\equiv 0 \\pmod{C_n(q)}, \\end{align*} where $0\\leqslant j\\leqslant 2r-1$. We also propose several related conjectures."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00003","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":"1703.00003","created_at":"2026-05-18T00:49:46.254048+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.00003v1","created_at":"2026-05-18T00:49:46.254048+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.00003","created_at":"2026-05-18T00:49:46.254048+00:00"},{"alias_kind":"pith_short_12","alias_value":"M7IGVK6PJFC5","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_16","alias_value":"M7IGVK6PJFC5LDXP","created_at":"2026-05-18T12:31:31.346846+00:00"},{"alias_kind":"pith_short_8","alias_value":"M7IGVK6P","created_at":"2026-05-18T12:31:31.346846+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/M7IGVK6PJFC5LDXPCKV2ECTCTR","json":"https://pith.science/pith/M7IGVK6PJFC5LDXPCKV2ECTCTR.json","graph_json":"https://pith.science/api/pith-number/M7IGVK6PJFC5LDXPCKV2ECTCTR/graph.json","events_json":"https://pith.science/api/pith-number/M7IGVK6PJFC5LDXPCKV2ECTCTR/events.json","paper":"https://pith.science/paper/M7IGVK6P"},"agent_actions":{"view_html":"https://pith.science/pith/M7IGVK6PJFC5LDXPCKV2ECTCTR","download_json":"https://pith.science/pith/M7IGVK6PJFC5LDXPCKV2ECTCTR.json","view_paper":"https://pith.science/paper/M7IGVK6P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.00003&json=true","fetch_graph":"https://pith.science/api/pith-number/M7IGVK6PJFC5LDXPCKV2ECTCTR/graph.json","fetch_events":"https://pith.science/api/pith-number/M7IGVK6PJFC5LDXPCKV2ECTCTR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/M7IGVK6PJFC5LDXPCKV2ECTCTR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/M7IGVK6PJFC5LDXPCKV2ECTCTR/action/storage_attestation","attest_author":"https://pith.science/pith/M7IGVK6PJFC5LDXPCKV2ECTCTR/action/author_attestation","sign_citation":"https://pith.science/pith/M7IGVK6PJFC5LDXPCKV2ECTCTR/action/citation_signature","submit_replication":"https://pith.science/pith/M7IGVK6PJFC5LDXPCKV2ECTCTR/action/replication_record"}},"created_at":"2026-05-18T00:49:46.254048+00:00","updated_at":"2026-05-18T00:49:46.254048+00:00"}