{"paper":{"title":"Properties of the Exceptional ($X_{\\ell}$) Laguerre and Jacobi Polynomials","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":["hep-th","math.CA","math.MP","nlin.SI"],"primary_cat":"math-ph","authors_text":"Choon-Lin Ho, Ryu Sasaki, Satoru Odake","submitted_at":"2009-12-30T11:02:06Z","abstract_excerpt":"We present various results on the properties of the four infinite sets of the exceptional $X_{\\ell}$ polynomials discovered recently by Odake and Sasaki [{\\it Phys. Lett. B} {\\bf 679} (2009), 414-417; {\\it Phys. Lett. B} {\\bf 684} (2010), 173-176]. These $X_{\\ell}$ polynomials are global solutions of second order Fuchsian differential equations with $\\ell+3$ regular singularities and their confluent limits. We derive equivalent but much simpler looking forms of the $X_{\\ell}$ polynomials. The other subjects discussed in detail are: factorisation of the Fuchsian differential operators, shape in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.5447","kind":"arxiv","version":4},"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"}