Dunkl operator

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1: 18.38 Mathematical Applications

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Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

►The Dunkl operator, introduced by Dunkl (1989), is an operator associated with reflection groups or root systems which has terms involving first order partial derivatives and reflection terms. Analogues of the original Dunkl operator (the rational case) were introduced by Heckman and Cherednik for the trigonometric case, and by Cherednik for the $q$-case. … ►In the one-variable case the Dunkl operator eigenvalue equation … …

2: Bibliography T

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Dunkl shift operators and Bannai-Ito polynomials. Adv. Math. 229 (4), pp. 2123–2158.

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External Links:

ISSN 0001-8708, Link, MathReview (Juan José Moreno-Balcázar), ZentralBlatt

Referenced by:

§18.28(xi).

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3: 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. …

4: Bibliography D

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Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1), pp. 167–183.

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External Links:

ISSN 0002-9947, Link, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§18.38(iii).

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5: 18.39 Applications in the Physical Sciences

… ►noting that the ${\psi }_{p,l}(r)$ are real, follows from the fact that the Schrödinger operator of (18.39.28) is self-adjoint, or from the direct derivation of Dunkl (2003). …

6: Bibliography R

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Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators. Academic Press, New York.

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External Links:

ISBN 0-12-585004-2(vol.4), MathReview (P. R. Chernoff)

Referenced by:

§1.18(x).

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Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii).

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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:

ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt

Referenced by:

§29.20(ii).

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On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (P. Saphar), ZentralBlatt

Referenced by:

§18.2(xii).

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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

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