Dunkl type operator

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1: 18.38 Mathematical Applications

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Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

… ►Algebraic structures were built of which special representations involve Dunkl type operators. …Eigenvalue equations involving Dunkl type operators have as eigenfunctions nonsymmetric analogues of multivariable special functions associated with root systems. … ►The Dunkl type operator is a $q$-difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial $R_n(z;a,b,c,d|q)$ and the ‘anti-symmetric’ Laurent polynomial $z^{-1}(1-az)(1-bz)R_{n-1}(z;qa,qb,c,d|q)$, where $R_n(z)$ is given in (18.28.1_5). … ►Dunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and $q$-Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7). …

2: 18.28 Askey–Wilson Class

… ►In Tsujimoto et al. (2012) an extension of the Bannai–Ito polynomials occurs as eigenfunctions of a Dunkl type operator. …

3: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

… ►The spaces ${𝒱}_n$ are eigenspaces of a second order partial differential operator, see (37.6.12). … ►

Rodrigues Type Formula

… ►The spaces ${𝒱}_n$ are eigenspaces of a second order partial differential operator: …

4: 37.4 Disk with Weight Function ${\left(1-x^2-y^2\right)}^{\alpha }$

… ►The spaces ${𝒱}_n^{\alpha }$ are eigenspaces of a second order partial differential operator, see (37.4.28). … ►See Dunkl and Xu (2014, §3.3.4) for multi-term relations satisfied by the OPs (37.4.5). … ►See Dunkl and Xu (2014, Proposition 2.3.6). … ►

Rodrigues type formula

… ►For these results see Appell and Kampé de Fériet (1926, pp. 263, 269), Erdélyi et al. (1953b, §§12.5, 12.6) (with $s=2\alpha +1$) and Dunkl and Xu (2014, §2.3) (with $\mu =\alpha +\frac{1}{2}$). …

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