A Bochner type characterization theorem for exceptional orthogonal polynomials

By M. García-Ferrero, D. Gómez‐Ullate, Robert Milson. Journal of Mathematical Analysis and Applications, 2019

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Abstract It was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be regarded as the extension of Bochner's result for classical orthogonal polynomials to the exceptional class. As a supplementary result, we derive a canonical form for exceptional operators based on a bilinear formalism, and prove that every exceptional operator has trivial monodromy at all primary poles.

Subfields: Applied Mathematics, Modeling and Simulation, Statistical and Nonlinear Physics

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