Non-Symmetric Heckman-Opdam Polynomials

• The paper introduces non-symmetric Heckman-Opdam polynomials as explicit multivariable orthogonal eigenfunctions defined via Cherednik-type differential–reflection operators.
• Shift operators and braid-type commutation relations are utilized to transition between polynomial degrees, generalizing classical Jacobi and Gegenbauer frameworks.
• Explicit squared-norms, orthogonality conditions, and sharp L^p-convergence ranges underpin the spectral theory and Fourier–Heckman–Opdam expansions.

Non-symmetric Heckman-Opdam polynomials are a distinguished class of explicit multivariable orthogonal polynomials (trigonometric, or rational, depending on context) characterized as joint eigenfunctions of Cherednik-type differential–reflection operators. These polynomials are indexed by integral weights and generalize the spectral theory of Jacobi-type differential operators and spherical harmonics. Their non-symmetric structure encodes the full double affine Hecke algebra (DAHA) symmetry and is crucial for Fourier analysis, explicit decomposition, and the construction of shift operators beyond the classical symmetric setting. The case of rank one—comprising types $A_1$ and $BC_1$—exhibits a complete and tractable structure that underpins much of the explicit analysis in this subject.

1. Definition and Explicit Structure

The non-symmetric Heckman-Opdam polynomials of type $A_1$, denoted $E_n^k(z)$ for $n\in\mathbb{Z}$ and multiplicity parameter $k\geq0$, are trigonometric polynomials characterized as unique solutions to the following conditions (Amri, 13 Jan 2026):

• Triangularity:

$$E_n^k(z) = e^{n z} + \sum_{j\triangleleft n} c_{n,j} e^{j z},$$

with the partial order $j\triangleleft n$ defined as $|j|<|n|$ with $|n|-|j|\in 2\mathbb{N}$, or $|j|=|n|$ and $j>n$.

• Orthogonality to Lower Terms:

$$(E_n^k(i x), e^{i j x})_k := \int_{-\pi}^\pi E_n^k(i x) e^{-i j x} dm_k(x) = 0$$

for all $j\triangleleft n$, with $dm_k(x) = |\sin x|^{2k} dx$.

• Cherednik Operator Eigenfunction:

For the operator

$$T^k f(x) = f'(x) + 2k \frac{f(x) - f(-x)}{1 - e^{-2x}},$$

one has

$$T^k(E_n^k) = n_k E_n^k, \qquad n_k = \begin{cases} n + k, & n > 0 \ n - k, & n \leq 0 \end{cases}$$

The set $\{ E_n^k(i x) \}_{n\in\mathbb{Z}}$ forms an orthogonal basis of $L^2([-\pi,\pi], dm_k)$, with explicit squared-norm

$$\|E_n^k\|_{2,k}^2 = \pi 2^{1-2k} n! \frac{\Gamma(n+2k+1)}{\Gamma(n+k+1)^2},$$

extended to $n<0$ via $\|E_{-n-1}^k\|_{2,k} = \|E_{n+1}^k\|_{2,k}$.

The associated symmetric polynomials, $P_n^k(x) = \frac{1}{2}(E_n^k(x) + E_n^k(-x))$, are even and solve the second-order Jacobi-type differential equation. On the unit circle, they are given by

$$P_n^k(i x) = \frac{\Gamma(k+1)}{\Gamma(2k+1)} \frac{n!}{\Gamma(n+k)} C_n^k(\cos x),$$

where $C_n^k$ denotes the Gegenbauer polynomial.

2. Shift Operators and the $q\to 1$ Limit from Askey–Wilson Theory

Shift operators play a central role in the theory of non-symmetric polynomials. In the Askey–Wilson (AW) regime, classified for the non-symmetric case in (Horssen et al., 2024), the shift operators are constructed as difference–reflection operators: $$\mathcal{G}^q_+,\, \mathcal{G}^q_-,\, \mathcal{E}^q_{1,\pm},\, \mathcal{E}^q_{2,\pm}$$ acting on the Laurent polynomial ring. Taking the formal limit $q\to1$ along $$(a,b,c,d)=(q^{k_1},-q^{k_2},q^{\frac12+k_3},-q^{\frac12+k_4})$$ yields first-order differential–reflection shift operators for BC$_1$–Heckman–Opdam polynomials.

For type $BC_1$, with $x=(z+z^{-1})/2$, $R f(x)=f(-x)$ the reflection operator:

• Forward shift: $$R_+ = \frac{z\partial_z}{z-z^{-1}} = \frac12 \partial_x$$
• Backward shift:

$$R_- = 2(x^2 - 1) \partial_x + 4k_1 x + 2k_3 - (x + \sqrt{x^2-1}R)$$

• Contiguous shifts: $E_{2,+}$ and $E_{2,-}$, also as explicit first-order differential–reflection operators

The shift algebra generated by these six operators satisfies braid-type commutation relations and intertwines the non-symmetric eigenbasis $\{E_n(x)\}$ of the rational Cherednik operator

$$\Omega = (x^2-1)\partial_x^2 + 2(k_1 + k_2)x\partial_x + 4k_3\partial_x - (k_1+\tfrac12)R.$$

On this basis,

$$R_+ E_n(x) = (n+k_1+k_2)E_{n-1}(x),\quad R_- E_n(x) = (n+k_1+k_2-1)E_{n+1}(x)$$

and similarly for $E_{2,\pm}$. On symmetric polynomials ($f(-x)=f(x)$), these reduce to classical symmetric Heckman–Opdam shift operators.

3. Norms, Orthogonality, and Spectral Data

The explicit squared-norms for the non-symmetric polynomials play a critical role in the analysis and spectral theory. For type $A_1$,

$$\|E_n^k\|_{2,k}^2 = \pi 2^{1-2k} n! \frac{\Gamma(n+2k+1)}{\Gamma(n+k+1)^2},\qquad n\geq0$$

with the symmetry $\|E_{-n-1}^k\|_{2,k} = \|E_{n+1}^k\|_{2,k}$. In type $BC_1$, the norm limit from Askey–Wilson theory yields

$$(E_n,E_n)_{HO} = \frac{2\Gamma(n+2k_1+2k_2)\Gamma(n+2k_3+1)}{n!\Gamma(2k_1+2k_2)\Gamma(2k_3+1)}$$

or in hypergeometric notation,

$$(E_n, E_n)_{HO} = \frac{2(k_1+k_2)_n\,(k_3+\tfrac12)_n}{n!\,(2k_1+2k_2)_n}$$

with $(\alpha)_n$ the Pochhammer symbol. Orthogonality follows from the explicit construction and spectral properties of the Cherednik operator.

4. Fourier–Heckman–Opdam Expansions and $L^p$-Convergence

Given $f\in L^p([-\pi,\pi], dm_k)$, the non-symmetric Fourier coefficients are

$$a_n = \gamma_n^2 \int_{-\pi}^\pi f(y)E_n^k(-i y)dm_k(y),\qquad \gamma_n = \|E_n^k\|_{2,k}^{-1}$$

and the partial sums

$S_N(f)(x) = \sum_{|n|\leq N} a_n E_n^k(i x).$

The partial sum operator has a kernel representation

$$S_N(f)(x) = \int_{-\pi}^\pi K_N(x,y) f(y) dm_k(y),\qquad K_N(x,y) = \sum_{|n|\leq N}\gamma_n^2 E_n^k(i x) E_n^k(-i y).$$

Utilizing explicit and telescoping formulas for $K_N(x,y)$, as well as kernel estimates and boundedness of weight-perturbed Hilbert transform operators, it is established that for every $k\geq0$ and every $f\in L^p([-\pi,\pi],dm_k)$ with

$2-\frac{1}{k+1} < p < 2+\frac{1}{k}$

the sequence $\{S_N(f)\}$ converges in the $L^p(dm_k)$ norm to $f$ (Amri, 13 Jan 2026). The argument hinges on explicit kernel decomposition, norm bounds, a duality argument, and the density of Heckman–Opdam polynomials in $L^p$ spaces. The sharp range of admissible $p$ is determined by the boundedness of specific weighted singular operators arising in the analysis.

5. Key Analytical and Algebraic Properties

Non-symmetric Heckman-Opdam polynomials exhibit several structural features:

• Shift operators: The explicit differential–reflection shifts (e.g., $R_\pm$, $E_{2,\pm}$) transmute between polynomials of different degrees and intertwine with the differential operator $\Omega$ (Horssen et al., 2024).
• Commutation relations: Braid-type relations hold among the shift operators, encoding the non-commutative algebraic structure.
• Transmutation identities: The shift operators satisfy

$$R_\pm \Omega = \Omega R_\pm, \quad E_{2,\pm}\Omega = \Omega E_{2,\pm}$$

ensuring the preservation of spectral type during polynomial degree shifts.

• Intertwining/Creation–Annihilation: Each shift operator acts as a creation or annihilation operator in the non-symmetric basis, paralleling the role of ladder operators in classical harmonic analysis.
• Symmetric restriction: On symmetric ($W_0$-invariant) polynomials, the shift algebra collapses to the familiar symmetric Jacobi–Heckman–Opdam case, thus recovering the standard theory.

6. Connections to Other Orthogonal Polynomial Systems and Outlook

The non-symmetric polynomials form the natural extension of symmetric (e.g., Jacobi, Gegenbauer) systems and are accessible via degenerations from $q$-Askey–Wilson theory. The explicit $q\to1$ limit provides bridge results between difference operators (AW theory) and differential–reflection operators (Heckman–Opdam theory). This cross-over enables the transfer of shift operators, norm formulas, and analytic tools from the $q$-deformed world to the trigonometric and rational regime.

A plausible implication is that the rich algebraic and analytic structure of the non-symmetric shift algebra for $BC_1$ generalizes to higher rank symmetric spaces and other root systems, facilitating explicit harmonic analysis, Plancherel theory, and potential applications to representation theory and special functions. The sharp $L^p$-convergence ranges for Fourier-type expansions in the non-symmetric basis offer a precise generalization of classical results to the double affine Hecke algebra framework.