{"paper":{"title":"Perturbed Hankel determinant, correlation functions and Painlev\\'e equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Engui Fan, Min Chen, Yang Chen","submitted_at":"2015-07-19T08:23:57Z","abstract_excerpt":"We continue with the study of the Hankel determinant, $$ D_{n}(t,\\alpha,\\beta):=\\det\\left(\\int_{0}^{1}x^{j+k}w(x;t,\\alpha,\\beta)dx\\right)_{j,k=0}^{n-1}, $$ generated by a Pollaczek-Jacobi type weight, $$ w(x;t,\\alpha,\\beta):=x^{\\alpha}(1-x)^{\\beta}{\\rm e}^{-t/x}, \\quad x\\in [0,1], \\quad \\alpha>0, \\quad \\beta>0, \\quad t\\geq 0. $$ This reduces to the \"pure\" Jacobi weight at $t=0.$ We may take $\\alpha\\in \\mathbb{R}$, in the situation while $t$ is strictly greater than $0.$ It was shown in Chen and Dai (2010), that the logarithmic derivative of this Hankel determinant satisfies a Jimbo-Miwa-Okamot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05261","kind":"arxiv","version":1},"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"}