A Bochner type classification theorem for exceptional orthogonal\n polynomials

2016/03/14 by María Ángeles García‐Ferrero, García-Ferrero, M. Ángeles, David Gómez‐Ullate +3 · 2 citations
Mathematics · #33C45 #33C47 #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical Inequalities and Applications #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1603.04358

openalex publication_date 2016/03/14 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

It was recently conjectured that every system of exceptional orthogonal\npolynomials is related to classical orthogonal polynomials by a sequence of\nDarboux transformations. In this paper we prove this conjecture, which paves\nthe road to a complete classification of all exceptional orthogonal\npolynomials. In some sense, this paper can be regarded as the extension of\nBochner's result for classical orthogonal polynomials to the exceptional class.\nAs a supplementary result, we derive a canonical form for exceptional operators\nbased on a bilinear formalism, and prove that every exceptional operator has\ntrivial monodromy at all primary poles.\n

Citations

• Note on the X_(1)-Jacobi orthogonal polynomials
• Über Sturm-Liouvillesche Polynomsysteme

Cited by

• The bispectral problem, the Darboux process, monodromy and the Hermite operator
• Asymptotics for Recurrence Coefficients of X1-Jacobi Polynomials and Christoffel Function

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