{"paper":{"title":"Discrete Painlev\\'e equations for recurrence coefficients of orthogonal polynomials","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Walter Van Assche","submitted_at":"2005-12-15T11:34:33Z","abstract_excerpt":"We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a discrete Painlev\\'e equation. The first example deals with Freud weights $|x|^\\rho \\exp(-|x|^m)$ on the real line, and we repeat Freud's derivation and analysis for the cases $m=2,4,6$. The Freud equations for the recurrence coefficients when $m=4$ corresponds to the discrete Painlev\\'e I equation. The second example deals with orthogonal polynomials on the unit circle for the weight $\\exp(\\lambda \\cos \\theta)$. These orthogonal polynomials are important in the theory of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0512358","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"}