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These identities may be expressed as $\\sum_{n\\ge 0}(1/p;1/q)_n= \\sum_{n\\ge 0} pq^n(p;q)_n(q;q)_n$ and $\\sum_{n\\ge 0} (-1)^n(1/p;1/q)_n= \\sum_{n\\ge 0} pq^n(p;q)_n(-q;q)_n =\\sum_{n\\ge 0} (q/p)^n(p;q^2)_n$, where the equalities apply to the (purely formal) power series expansions of the above expressions at $p=q=1$, as well as at other suitable roots of unity."},"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":"1309.6669","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-25T21:15:42Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"7d0aade76aca0577881bf9202e91fa4bc3f6bb7af264b84d3f2003bed4d78098","abstract_canon_sha256":"7b432befe984ebd5e5935c6b755d436d6c1951fd6473f33f090ba792e30f24d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:10.948496Z","signature_b64":"75mqeuHdRRHXkUuBHAgFNsfxCshc7OpElfDwvPAhT/nVUvmQDvbeAf+a/hh6xtsUSDlAvv5oeMqVw5ITWS8mDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64d85b1e377a44c6bca40ed7e799c96f5f7015558641e8bae4029d34acd45d44","last_reissued_at":"2026-05-18T03:12:10.947660Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:10.947660Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On q-Series Identities Related to Interval Orders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"George E. Andrews, V\\'it Jel\\'inek","submitted_at":"2013-09-25T21:15:42Z","abstract_excerpt":"We prove several power series identities involving the refined generating function of interval orders, as well as the refined generating function of the self-dual interval orders. These identities may be expressed as $\\sum_{n\\ge 0}(1/p;1/q)_n= \\sum_{n\\ge 0} pq^n(p;q)_n(q;q)_n$ and $\\sum_{n\\ge 0} (-1)^n(1/p;1/q)_n= \\sum_{n\\ge 0} pq^n(p;q)_n(-q;q)_n =\\sum_{n\\ge 0} (q/p)^n(p;q^2)_n$, where the equalities apply to the (purely formal) power series expansions of the above expressions at $p=q=1$, as well as at other suitable roots of unity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6669","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":"1309.6669","created_at":"2026-05-18T03:12:10.947805+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.6669v1","created_at":"2026-05-18T03:12:10.947805+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6669","created_at":"2026-05-18T03:12:10.947805+00:00"},{"alias_kind":"pith_short_12","alias_value":"MTMFWHRXPJCM","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_16","alias_value":"MTMFWHRXPJCMNPFE","created_at":"2026-05-18T12:27:52.871228+00:00"},{"alias_kind":"pith_short_8","alias_value":"MTMFWHRX","created_at":"2026-05-18T12:27:52.871228+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/MTMFWHRXPJCMNPFEB3L6PGOJN5","json":"https://pith.science/pith/MTMFWHRXPJCMNPFEB3L6PGOJN5.json","graph_json":"https://pith.science/api/pith-number/MTMFWHRXPJCMNPFEB3L6PGOJN5/graph.json","events_json":"https://pith.science/api/pith-number/MTMFWHRXPJCMNPFEB3L6PGOJN5/events.json","paper":"https://pith.science/paper/MTMFWHRX"},"agent_actions":{"view_html":"https://pith.science/pith/MTMFWHRXPJCMNPFEB3L6PGOJN5","download_json":"https://pith.science/pith/MTMFWHRXPJCMNPFEB3L6PGOJN5.json","view_paper":"https://pith.science/paper/MTMFWHRX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.6669&json=true","fetch_graph":"https://pith.science/api/pith-number/MTMFWHRXPJCMNPFEB3L6PGOJN5/graph.json","fetch_events":"https://pith.science/api/pith-number/MTMFWHRXPJCMNPFEB3L6PGOJN5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MTMFWHRXPJCMNPFEB3L6PGOJN5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MTMFWHRXPJCMNPFEB3L6PGOJN5/action/storage_attestation","attest_author":"https://pith.science/pith/MTMFWHRXPJCMNPFEB3L6PGOJN5/action/author_attestation","sign_citation":"https://pith.science/pith/MTMFWHRXPJCMNPFEB3L6PGOJN5/action/citation_signature","submit_replication":"https://pith.science/pith/MTMFWHRXPJCMNPFEB3L6PGOJN5/action/replication_record"}},"created_at":"2026-05-18T03:12:10.947805+00:00","updated_at":"2026-05-18T03:12:10.947805+00:00"}