{"paper":{"title":"Evaluations of some Toeplitz-type determinants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Han Wang, Zhi-Wei Sun","submitted_at":"2022-06-24T14:32:23Z","abstract_excerpt":"In this paper we evaluate some Toeplitz-type determinants. Let $n>1$ be an integer. We prove the following two basic identities: \\begin{align*} \\det{[j-k+\\delta_{jk}]_{1\\leq j,k\\leq n}}&=1+\\frac{n^2(n^2-1)}{12}, \\\\ \\det{[|j-k|+\\delta_{jk}]_{1\\leq j,k\\leq n}}&= \\begin{cases} \\frac{1+(-1)^{(n-1)/2}n}{2}&\\text{if}\\ 2\\nmid n,\\\\ \\frac{1+(-1)^{n/2}}{2}&\\text{if}\\ 2\\mid n, \\end{cases} \\end{align*} where $\\delta_{jk}$ is the Kronecker delta. For complex numbers $a,b,c$ with $b\\not=0$ and $a^2\\not=4b$, and the sequence $(w_m)_{m\\in\\mathbb Z}$ with $w_{k+1}=aw_k-bw_{k-1}$ for all $k\\in\\mathbb Z$, we est"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.12317","kind":"arxiv","version":8},"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/2206.12317/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"}