{"paper":{"title":"On determinants involving second-order recurrent sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2023-02-16T14:20:34Z","abstract_excerpt":"Let $A$ and $B$ be complex numbers, and let $(w_n)_{n\\ge0}$ be a sequence of complex numbers with $w_{n+1}=Aw_n-Bw_{n-1}$ for all $n=1,2,3,\\ldots$. When $w_0=0$ and $w_1=1$, the sequence $(w_n)_{n\\ge0}$ is just the Lucas sequence $(u_n(A,B))_{n\\ge0}$. In this paper, we evaluate the determinants $$\\det[w_{|j-k|}]_{1\\le j,k\\le n}\\ \\ \\text{and}\\ \\ \\det[w_{|j-k+1|}]_{1\\le j,k\\le n}.$$ In particular, we have $$\\det[u_{|j-k|}(A,B)]_{1\\le j,k\\le n}=(-1)^{n-1}u_{n-1}(2A,(B+1)^2).$$ When $B=-1$ and $2\\mid n$, we also determine the characteristic polynomial of the matrix $[w_{j+k}]_{0\\le j,k\\le n-1}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.08315","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2302.08315/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}