{"paper":{"title":"Discrete Entropy of Generalized Jacobi Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ana Pe\\~na, Andrei Martinez-Finkelshtein, Paul Nevai","submitted_at":"2014-10-08T21:26:48Z","abstract_excerpt":"Given a sequence of orthonormal polynomials on $\\Bbb R$,$\\{p_n\\}_{n\\geq 0}$, with $p_n$ of degree $n$, we define the discrete probability distribution $\\Psi_n(x) = \\left(\\Psi_{n,1}(x), \\dots \\Psi_{n,n}(x) \\right) $, with $\\Psi_{n,j}(x) = \\big(\\sum_{j=0}^{n-1} p_j^2(x)\\big)^{-1} p_{j-1}^2(x)$, $j=1, \\dots, n$. In this paper, we study the asymptotic behavior as $n\\to \\infty$ of the Shannon entropy $\\mathcal S ((\\Psi_n(x))= -\\sum_{j=1}^n \\Psi_{n,j}(x) \\log (\\Psi_{n,j}(x))$, $x\\in (-1,1)$, when the orthogonality weight is $ (1-x)^{\\alpha}\\, (1+x)^{\\beta}\\, h(x) $, $\\alpha, \\beta > -1$, and where $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.2286","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":""},"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"}