Exceptional Orthogonal Polynomials Overview

• Exceptional orthogonal polynomials are complete orthogonal systems with deliberate gaps in their degree sequence, breaking classical assumptions.
• They are constructed through rational Darboux transformations, transforming classical operators into novel families with enriched spectral properties.
• These polynomials find practical applications in quantum models, spectral theory, and exactly solvable differential equations.

Exceptional orthogonal polynomials (EOPs) are families of real or complex polynomials that form complete orthogonal systems with respect to positive measures and are eigenfunctions of second-order differential (or difference) operators, yet differ from the classical (Hermite, Laguerre, Jacobi, etc.) families by the presence of "gaps"—certain degrees are missing from the sequence. These missing degrees, or "codimension," break the assumption underpinning Bochner’s theorem, thereby yielding new and much richer families with applications in mathematical physics, exactly solvable quantum models, and spectral theory.

1. Defining Features and Classification of Exceptional Orthogonal Polynomials

EOPs are characterized by sequences $\{p_n\}_{n\in\mathbb{N}_0\setminus A}$, where $A$ is a finite subset of the non-negative integers, so that $\deg p_n = n$ for all $n \in \mathbb{N}_0 \setminus A$; that is, not every degree occurs. Despite these gaps, EOPs maintain orthogonality and completeness in the associated $L^2$ space. The classical families (Hermite, Laguerre, Jacobi) can be viewed as the codimension-zero case (no gaps, $A = \varnothing$).

The classification of EOPs has now achieved a "Bochner-type" completeness: every such system arises from a finite sequence of Darboux (or rational isospectral) transformations applied to a classical operator (Gomez-Ullate et al., 2011, García-Ferrero et al., 2016). The differential operator always has polynomial leading coefficient (degree exactly 2), and lower-order coefficients are rational or meromorphic functions whose poles and associated "gap multiplicities" encode the degrees omitted from the EOP sequence. Every EOP system is thus Darboux-connected to its classical ancestor.

2. Construction: Rational Factorizations and Darboux Transformations

A foundational mechanism underlying exceptional orthogonal polynomials is the rational (or algebraic) factorization of second-order differential operators, which is a generalization of the classical Darboux transformation (Gomez-Ullate et al., 2011, Gomez-Ullate et al., 2011). For a generic operator

$T(y) = p(x) y'' + q(x) y' + r(x) y$

with a quasi-rational eigenfunction $\phi(x)$ (i.e., $(\phi'/\phi)$ rational), $T$ can be factorized as $T = B A + \lambda_0$ with first-order operators $A$ and $B$ related to $\phi$. Iterating this construction (multi-step Darboux/Crum chain) using sequences of quasi-rational eigenfunctions with distinct spectral parameters yields higher-codimension EOPs and partner operators whose polynomial flags begin at higher degree.

A prototypical example is the "two-step" Darboux transformation yielding exceptional Laguerre polynomials of type II (Gomez-Ullate et al., 2011). Here, two quasi-rational seed functions associated with transformed Laguerre equations are used to build a Wronskian structure: $$I_n(z) = Cz^{-k} W[\varphi_1(z), \varphi_2(z), L_{n-\ell}^{(k+2)}(z)]$$ for $n \geq \ell$, where $\ell$ is determined by the degrees of the seed polynomials. The new EOPs satisfy a second-order eigenvalue equation and exhibit gaps determined by the step sequence.

3. Operator Properties, Orthogonality, and Weights

EOPs are eigenfunctions of second-order differential operators with rational coefficients: $$T^{(EOP)}y(x) = \lambda_n y(x),\qquad n\in \mathbb{N}_0\setminus A$$ and are orthogonal with respect to a weight function $W(x)$ that is the classical weight divided by a squared polynomial (the "deformation polynomial" or "denominator"), whose zeros correspond precisely to the "gaps". For Jacobi-type EOPs (codimension $m$), this takes the form (Gomez-Ullate et al., 2011): $$\widehat{W}_{a,B,m}(x) = \frac{W_{a+1,B+1}(x)}{S_{a+1,B-1,m}(x)^2}$$ with $W_{a+1,B+1}$ the classical Jacobi weight and $S_{a+1,B-1,m}(x)$ a degree-$m$ polynomial determined by the Darboux construction. The admissibility of the parameter set $(a,B)$ is determined by the requirement that $S_{a+1,B-1,m}$ has no zeros in the orthogonality interval, ensuring the nonsingularity of $W$ and the completeness of the EOP family.

Analogous constructions hold for Laguerre and Hermite types; for instance, exceptional Hermite polynomials arise as Wronskians of classical Hermite polynomials and are orthogonal with respect to $e^{-x^2}/[Q(x)]^2$ where $Q(x)$ is the deformed denominator (Duran, 2013, Haese-Hill et al., 2015).

4. Flag Theory, Admissibility, and Determinantal Structures

EOPs depart from standard polynomial flags by starting at degree $m > 0$. Nonstandard polynomial flags of codimension $m$ are classified by the preservation properties under the exceptional operator (Gomez-Ullate et al., 2011). The most general codimension-one flag is, up to affine transformation, $\{x+1, x^2, x^3, \ldots\}$, and its existence is tightly linked to the non-polynomial nature of the preserving operator.

Determinantal formulas, notably Wronskian and Casorati determinants, are central to defining and analyzing EOPs. For finite sets $F$ or pairs $F_1,F_2$ indicating the degrees to be deleted, exceptional Charlier, Meixner, Hermite, and Laguerre polynomials can be explicitly constructed as determinants whose entries are the corresponding classical polynomials evaluated at shifted or derived arguments (Duran, 2013, Duran, 2013, Simanek, 2022). Admissibility conditions—essentially, positivity or nonvanishing of certain determinant structures over the orthogonality support—are critical to ensure a positive orthogonality measure and thus the existence of a complete orthogonal system.

5. Spectral Theory, Completeness, and Applications

The self-adjoint spectral theory of the differential operators associated with EOPs is well developed (Liaw et al., 2014). The maximal domain of the exceptional operator generally involves singular endpoints, and boundary conditions (e.g., limit-circle at $x=0$ for certain parameter ranges) are required to render the operator self-adjoint. The spectrum of the exceptional operator consists of a pure point spectrum corresponding to the sequence $\{n\in\mathbb{N}_0\setminus A\}$, and the exceptional polynomials provide a complete orthogonal basis in the associated weighted $L^2$ space.

EOPs have become essential in constructing exactly solvable quantum-mechanical potentials (via SUSY and isospectral deformation), in the analysis of Fokker–Planck/diffusion processes (Chou et al., 2012), and in renewal of bispectrality and Darboux-commutativity properties (Castro et al., 2022).

6. Interrelations, Asymptotics, and Transformations

Asymptotic and limit relations interconnect the EOP families of Jacobi, Laguerre, and Hermite type, generalizing classical limit theorems. For example, $X_m$-Laguerre EOPs (of any type) can be obtained as scaling limits of $X_m$-Jacobi EOPs of the same type, and Hermite type-III EOPs are limits of Jacobi or Laguerre type-III EOPs (Quesne, 2023). Quadratic transformations relating classical Hermite and Laguerre polynomials extend naturally to even-codimension Hermite EOPs.

The zeros of exceptional orthogonal polynomials exhibit new behaviors; so-called "exceptional zeros" lie outside the classical orthogonality support and converge at rates depending on the family (O($n^{-1}$) Jacobi; O($n^{-1/2}$) Laguerre/Hermite) to well-defined limit points as the degree grows (Simanek, 2020).

7. Determinantal and Moment Representations

Determinantal representations, closely connected to Vandermonde structures and the zeros of classical polynomials, provide efficient formulas for $X_m$-Laguerre, $X_m$-Jacobi, and higher codimension Hermite EOPs (Simanek, 2022). Universal moment representations for $X_1$ EOPs use "adjusted moments" to encode the orthogonality and exceptional constraints in a single determinant, with generalization prospects to high-codimension systems (Liaw et al., 2016).

Table: Comparison of Classical vs. Exceptional Orthogonal Polynomials

Aspect	Classical OPs	Exceptional OPs
Degree sequence	All $n \geq 0$	$n \in \mathbb{N}_0 \setminus A$, $A$ finite omissions
Leading coefficient of OP	$p_n(x)$ monic/prescribed	$\deg p_n = n$, but lowest $n$ omitted
Operator coefficients	Polynomial	Rational (poles at exceptional points)
Orthogonality measure	Classical weight	Classical weight divided by polynomial squared
Recurrence	Three-term	Higher-order, reflects codimension
Spectral theory	Pure point, full completeness	Pure point, completeness for admissible weights
Transformation origin	None	Darboux/Crum transformations from classical

• "On orthogonal polynomials spanning a non-standard flag" (Gomez-Ullate et al., 2011)
• "Two-step Darboux transformations and exceptional Laguerre polynomials" (Gomez-Ullate et al., 2011)
• "A Bochner type classification theorem for exceptional orthogonal polynomials" (García-Ferrero et al., 2016)
• "A New Class of Exceptional Orthogonal Polynomials: The Type III $X_{m}$-Laguerre Polynomials..." (Liaw et al., 2014)
• "Exceptional orthogonal polynomials and generalized Schur polynomials" (Grandati, 2013)
• "Exceptional Charlier and Hermite orthogonal polynomials" (Duran, 2013)
• "Exceptional Meixner and Laguerre orthogonal polynomials" (Duran, 2013)
• "Convergence Rates of Exceptional Zeros of Exceptional Orthogonal Polynomials" (Simanek, 2020)
• "Determintal Formulas for Exceptional Orthogonal Polynomials" (Simanek, 2022)
• "Connecting Exceptional Orthogonal Polynomials of Different Kind" (Quesne, 2023)

This synthesis provides the essential definitions, constructions, operator-theoretic framework, and structural features central to exceptional orthogonal polynomials, situating them as Darboux-deformations of the classical theory with deep applications in spectral and mathematical physics.