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Define q-Euler numbers $E_n(q)$, q-Sali\\'e numbers $S_n(q)$ and q-Carlitz numbers $C_n(q)$ as follows:\n  $$\\sum_{n=0}^{\\infty}E_n(q)\\frac{x^n}{(q,q)_n} =1/\\sum_{n=0}^{\\infty}\\frac{q^{n(2n-1)}x^{2n}}{(q;q)_{2n}},$$\n  $$\\sum_{n=0}^{\\infty}S_n(q)\\frac{x^n}{(q;q)_n} =\\sum_{n=0}^{\\infty}\\frac{q^{n(n-1)}x^{2n}}{(q;q)_{2n}} /\\sum_{n=0}^{\\infty}\\frac{(-1)^nq^{n(2n-1)}x^{2n}}{(q;q)_{2n}},$$\n  $$\\sum_{n=0}^{\\infty}C_n(q)\\frac{x^n}{(q;q)_n} =\\sum_{n=0}^{\\infty}\\frac{q^{n(n-1)}x^{2n+1}}{(q;q)_{2n+1}} /\\sum_{n=0}^{\\infty}\\frac{(-1)^nq^{n(2n+1)}x^{2n+1"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0505548","kind":"arxiv","version":6},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2005-05-25T18:15:54Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"2b55436ba4c06c0a4f3d087e5658f29fa4e6b31c25c60d81823475ffbd47bf9b","abstract_canon_sha256":"a585e62d3ecea71542ac4f531513f07ecac6023e9da633b104479edeecf7ff3d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:24.979290Z","signature_b64":"8vPKoGZI9vOL2a9NGW/GJjkYbwYC+tPVXLZhnArrShng8EmxeuJXRcAr+JD4z9OQldsH6mnRYrznd+fvJVESBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ec5c5273c3e5a564eaaebdb2152c0cb1eabe83e9e4b818df2b8e03fce22200d","last_reissued_at":"2026-05-18T01:38:24.978370Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:24.978370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On q-Euler numbers, q-Salie numbers and q-Carlitz numbers","license":"","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Hao Pan, Zhi-Wei Sun","submitted_at":"2005-05-25T18:15:54Z","abstract_excerpt":"Let $(a;q)_n=\\prod_{0\\le k<n}(1-aq^k)$ for n=0,1,2,.... Define q-Euler numbers $E_n(q)$, q-Sali\\'e numbers $S_n(q)$ and q-Carlitz numbers $C_n(q)$ as follows:\n  $$\\sum_{n=0}^{\\infty}E_n(q)\\frac{x^n}{(q,q)_n} =1/\\sum_{n=0}^{\\infty}\\frac{q^{n(2n-1)}x^{2n}}{(q;q)_{2n}},$$\n  $$\\sum_{n=0}^{\\infty}S_n(q)\\frac{x^n}{(q;q)_n} =\\sum_{n=0}^{\\infty}\\frac{q^{n(n-1)}x^{2n}}{(q;q)_{2n}} /\\sum_{n=0}^{\\infty}\\frac{(-1)^nq^{n(2n-1)}x^{2n}}{(q;q)_{2n}},$$\n  $$\\sum_{n=0}^{\\infty}C_n(q)\\frac{x^n}{(q;q)_n} =\\sum_{n=0}^{\\infty}\\frac{q^{n(n-1)}x^{2n+1}}{(q;q)_{2n+1}} /\\sum_{n=0}^{\\infty}\\frac{(-1)^nq^{n(2n+1)}x^{2n+1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0505548","kind":"arxiv","version":6},"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/0505548","created_at":"2026-05-18T01:38:24.978556+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0505548v6","created_at":"2026-05-18T01:38:24.978556+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0505548","created_at":"2026-05-18T01:38:24.978556+00:00"},{"alias_kind":"pith_short_12","alias_value":"J3C4KJZ4HZNF","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_16","alias_value":"J3C4KJZ4HZNFMTVK","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_8","alias_value":"J3C4KJZ4","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/J3C4KJZ4HZNFMTVK5PNSCUWAZM","json":"https://pith.science/pith/J3C4KJZ4HZNFMTVK5PNSCUWAZM.json","graph_json":"https://pith.science/api/pith-number/J3C4KJZ4HZNFMTVK5PNSCUWAZM/graph.json","events_json":"https://pith.science/api/pith-number/J3C4KJZ4HZNFMTVK5PNSCUWAZM/events.json","paper":"https://pith.science/paper/J3C4KJZ4"},"agent_actions":{"view_html":"https://pith.science/pith/J3C4KJZ4HZNFMTVK5PNSCUWAZM","download_json":"https://pith.science/pith/J3C4KJZ4HZNFMTVK5PNSCUWAZM.json","view_paper":"https://pith.science/paper/J3C4KJZ4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0505548&json=true","fetch_graph":"https://pith.science/api/pith-number/J3C4KJZ4HZNFMTVK5PNSCUWAZM/graph.json","fetch_events":"https://pith.science/api/pith-number/J3C4KJZ4HZNFMTVK5PNSCUWAZM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J3C4KJZ4HZNFMTVK5PNSCUWAZM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J3C4KJZ4HZNFMTVK5PNSCUWAZM/action/storage_attestation","attest_author":"https://pith.science/pith/J3C4KJZ4HZNFMTVK5PNSCUWAZM/action/author_attestation","sign_citation":"https://pith.science/pith/J3C4KJZ4HZNFMTVK5PNSCUWAZM/action/citation_signature","submit_replication":"https://pith.science/pith/J3C4KJZ4HZNFMTVK5PNSCUWAZM/action/replication_record"}},"created_at":"2026-05-18T01:38:24.978556+00:00","updated_at":"2026-05-18T01:38:24.978556+00:00"}