A Bochner type classification theorem for exceptional orthogonal polynomials

Abstract

It was recently conjectured that every system of exceptional orthogonal polynomials is related to classical orthogonal polynomials 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.

a number of minor mistakes have been corrected. some proofs have been streamlined and simplified

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Cited by in corpus (10)

• Exceptional Jacobi polynomials
• Exceptional Laguerre polynomials
• Exceptional Legendre Polynomials and Confluent Darboux Transformations
• The Matrix Bochner Problem
• Moment Representations of Exceptional $X_1$ Orthogonal Polynomials
• Note on the Equilibrium Measures of Julia sets of Exceptional Jacobi Polynomials
• Translation operator with exceptional Laguerre polynomials
• Discrete diffusion semigroups associated with Dunkl-Jacobi and exceptional Jacobi polynomials
• QHJ route to multi-indexed exceptional Laguerre polynomials and corresponding rational potentials
• Asymptotics for Recurrence Coefficients of X1-Jacobi Polynomials and Christoffel Function