Dunkl-Classical Orthogonal Polynomials

• Dunkl-classical orthogonal polynomials are eigenfunctions of first-order Dunkl-type differential-difference operators that incorporate reflection symmetry.
• They are uniquely characterized as the big -1 Jacobi and little -1 Jacobi families, emerging as q = -1 limits of the q-Jacobi polynomial systems.
• Their classification relies on strict symmetrizability conditions, positive weight measures, and Pearson-type differential equations ensuring orthogonality.

The Dunkl-classical orthogonal polynomials form a distinguished subclass of orthogonal polynomial systems characterized by the fact that they are eigenfunctions of first-order differential-difference operators of Dunkl type, i.e., operators that involve both differentiation and reflection. This class is a direct analogue—yet a strict reduction—of the classical families singled out by Bochner’s theorem, and their classification is governed by a Bochner-type theorem adapted to the Dunkl context. The central result is that, up to affine transformations and limiting cases, only the so-called big $-1$ Jacobi and little $-1$ Jacobi polynomials admit such a Dunkl-classical structure at the first-order level. These are realized as $q=-1$ limits of the big and little $q$-Jacobi polynomials. This establishes a precise basis for understanding classicality in the presence of reflection symmetry as encoded by Dunkl operators.

1. Dunkl-Type Operators and Polynomial Eigenfunction Paradigm

A Dunkl-type operator $L$ on the line is a first-order linear differential-difference operator that acts on smooth functions $f(x)$ via

$$L f(x) = F_0(x) f(x) + F_1(x) f(-x) + G_0(x) \frac{d}{dx} f(x) + G_1(x) \frac{d}{dx} f(-x)$$

with $F_0, F_1, G_0, G_1$ rational functions of $x$. The reflection operator $R$ is defined by $R f(x) = f(-x)$. Such an operator generalizes classical differential operators by incorporating reflection symmetry.

The "Dunkl-classical" property is defined by the existence of a sequence of monic orthogonal polynomials $\{P_n(x)\}_{n\ge0}$, each of exact degree $n$, that are eigenfunctions of $L$: $L P_n(x) = \lambda_n P_n(x)$ for nondegenerate eigenvalues $\lambda_n$.

A crucial additional requirement is symmetrizability: there must exist a positive weight $w(x)$ on a symmetric interval (or symmetric unions) such that $w(x) L$ is formally self-adjoint. This is a nontrivial constraint in the Dunkl setting and restricts the operator’s admissible form.

2. Bochner-Type Classification for First-Order Dunkl Operators

The central theorem, an analogue of Bochner’s original result, classifies all possible first-order Dunkl-type operators with a complete family of polynomial eigenfunctions and orthogonality with respect to a positive weight as follows:

Theorem (Dunkl-Bochner Analogue):

Under the above conditions, the only orthogonal polynomial families $\{P_n(x)\}$ (for an infinite sequence, not terminating after a finite degree) that are eigenfunctions of a symmetrizable, first-order Dunkl-type operator are:

• The big $-1$ Jacobi polynomials;
• The little $-1$ Jacobi polynomials (as a limiting case).

Explicitly, the operator simplifies to the form: $L = F(x)(I - R) + G_1(x)\partial_x R$ with $F(x)$, $G_1(x)$ as rational functions (parameters detailed below), and $G_0(x) = 0$ after normalization.

Parameterizations

The two classes correspond to specific forms:

• Big $-1$ Jacobi:

\begin{align*} G_1(x) &= \frac{2(x-1)(x+c)}{x}, \ F(x) &= -\frac{c}{x^2} + \frac{\beta - \alpha c}{x} - (\alpha + \beta + 1) \end{align*} with real parameters $\alpha > -1$, $\beta > -1$, $0 < c < 1$.

• Little $-1$ Jacobi (limit $c \to 0$):

\begin{align*} G_1(x) &= 2(1-x), \ F(x) &= \alpha + \beta + 1 - \frac{\beta}{x} \end{align*}

No other choices of $F(x), G_1(x)$ yield a symmetrizable operator with a positive orthogonality measure and an infinite sequence of polynomial eigenfunctions; in particular, no classical Hermite or Laguerre analogues arise at this level.

3. Orthogonality Measures and Supports

The identification of classicality in the Dunkl sense is tightly coupled to the existence of a positive weight function for orthogonality. The explicit weights and supports are:

Operator	Polynomial Family	Weight Function	Support
$L$ with $G_1$, $F$	Big $-1$ Jacobi	$$\dfrac{|x|}{x}(x+1)(x-c)(1-x^2)^{\frac{\alpha-1}{2}}(x^2-c^2)^{\frac{\beta-1}{2}}$$	$[-1, -c]\cup [c, 1]$
$L$ with $c = 0$	Little $-1$ Jacobi	$(x+1)(1-x^2)^{(\alpha-1)/2}|x|^{\beta}$	$[-1, 1]$

Each weight function $w(x)$ further satisfies a Pearson-type differential equation tailored to the symmetrizability condition. For big $-1$ Jacobi, this becomes (for $y = x^2$): $$2(y-1)(y-c^2) W'(y) + [(2-\alpha-\beta)y + \beta - 1 + c^2(\alpha-1)] W(y) = 0$$ where $w(x) = \theta(x)(x+1)(x-c)W(x^2)$.

4. Relation to $q$-Jacobi Polynomials and Limiting Process

The big $-1$ and little $-1$ Jacobi polynomials arise as the $q \to -1$ limit in the Askey $q$-Jacobi hierarchy:

• Construction: These polynomials are precisely the limits of the big and little $q$-Jacobi polynomials as $q \to -1$, with parameterizations naturally fixing the limiting process.
• Preservation of classical-like structure: Despite being governed by first-order Dunkl-type operators, these families retain properties such as orthogonality and explicit three-term recurrences.
• Implications: This link exhausts all possible infinite, positive-definite families for first-order Dunkl operator eigenproblems, paralleling the exhaustiveness of Bochner's theorem for second-order operators.

5. Degeneracies and Excluded Cases

Systematic analysis of degenerate parameter choices confirms that no further (nontrivial) orthogonal polynomial families appear with the required symmetrizability. In particular:

• There are no Dunkl analogues of Hermite or Laguerre polynomials as first-order Dunkl operator eigenfunctions with positive-definite weights.
• Generalized Hermite and Gegenbauer polynomials do appear as "Dunkl-classical," but only as eigenfunctions of higher-order Dunkl-type differential-difference operators, e.g., those of the form

$L_\mu = \sigma(x) T^2_\mu + \tau(x) T_\mu$

where $T_\mu$ is the Dunkl operator.

6. Context Within Bochner-Type Theory and Significance

• The Dunkl-Bochner classification confirms a drastic narrowing of the possible families compared to the classical (second-order differential) or $q$-difference contexts.
• The identified families (big and little $-1$ Jacobi) correspond to the $q=-1$ endpoints of the $q$-Askey scheme, a position reflecting the highest possible generality compatible with first-order Dunkl symmetry.
• The result underscores the uniqueness of "Dunkl-classical" orthogonal polynomials at the first-order level, reinforcing the centrality of reflection-symmetry in the classification.

7. Summary Table of Operators, Polynomials, and Weights

Operator Form	Polynomial Family	Weight Function	Support
$L = F(x)(I - R) + G_1(x)\partial_x R$	Big $-1$ Jacobi	$$\frac{|x|}{x}(x+1)(x-c)(1-x^2)^{(\alpha-1)/2}(x^2-c^2)^{(\beta-1)/2}$$	$[-1,-c] \cup [c,1]$
$L$ with $c \to 0$	Little $-1$ Jacobi	$(x+1)(1-x^2)^{(\alpha-1)/2}|x|^\beta$	$[-1, 1]$

References to Principal Formulas and Theorems

• General operator structure: $L$ formulas as above
• Eigenproblem: $L P_n(x) = \lambda_n P_n(x)$
• Classification theorem: uniqueness asserted after enforcing symmetrizability and positivity (see above forms and parameterizations)
• Pearson equation for orthogonality measure: as detailed for big $-1$ Jacobi

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The classification of Dunkl-classical orthogonal polynomials at the first-order operator level is now complete: these systems are limited to the big and little $-1$ Jacobi families, arising as $q=-1$ limits of the corresponding $q$-Jacobi polynomials. This outcome precisely mirrors the spirit of Bochner's theorem and sets the standard for "classicality" in reflection-invariant operator-theoretic settings (Vinet et al., 2010).