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.

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

References in corpus (21)

• An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
• An Extension of Bochner's Problem: Exceptional Invariant Subspaces
• Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
• Two-step Darboux transformations and exceptional Laguerre polynomials
• Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics
• Solvable rational extensions of the isotonic oscillator
• Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials
• Asymptotic behaviour of zeros of exceptional Jacobi and Laguerre polynomials
• Extended Krein-Adler theorem for the translationally shape invariant potentials
• Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators
• On Stieltjes relations, Painlevé-IV hierarchy and complex monodromy
• Combined state-adding and state-deleting approaches to type III multi-step rationally-extended potentials: applications to ladder operators and superintegrability
• A new recurrence formula for generic exceptional orthogonal polynomials
• The Scattering amplitude for one parameter family of shape invariant potentials related to Xm Jacobi polynomials
• The Scattering amplitude for Rationally extended shape invariant Eckart potentials
• Multi-indexed Jacobi polynomials and Maya diagrams
• Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity
• Spectral analysis for the exceptional $X_m$-Jacobi equation
• A New Class of Exceptional Orthogonal Polynomials: The Type III $X_m$-Laguerre Polynomials And The Spectral Analysis of Three Types of Exceptional Laguerre Polynomials
• Note on the X_(1)-Jacobi orthogonal polynomials
• Note on the X_(1)-Laguerre orthogonal polynomials

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