{"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"}