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. … ►In the one-variable case the Dunkl operator eigenvalue equation … …

2: Bibliography D

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Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-80043-9, MathReview (Marcel de Jeu), ZentralBlatt

Referenced by:

§18.37(ii), §18.37(i).

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

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

Referenced by:

§18.38(iii).

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A Laguerre polynomial orthogonality and the hydrogen atom. Anal. Appl. (Singap.) 1 (2), pp. 177–188.

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ISSN 0219-5305, Document, Link, MathReview (E. Kreyszig), ZentralBlatt

Referenced by:

§18.39(ii), (33.14.15), Erratum (V1.0.24) for Equation (33.14.15).

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Special Functions. World Scientific Publishing Co., Inc., River Edge, NJ.

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External Links:

ISBN 981-02-4393-6, Document, Link, MathReview Entry

Referenced by:

Profile Mourad E.H. Ismail.

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Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.

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External Links:

Document, MathReview (R. C. Varma), ZentralBlatt

Referenced by:

§18.17(iii).

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3: Bibliography T

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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

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Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15 (2), pp. 207–225.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§10.19(iii), §10.20(i), §10.74(i).

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The universal Askey-Wilson algebra and DAHA of type $(C_1^{\vee },C_1)$ . SIGMA 9, pp. Paper 047, 40 pp..

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ISSN 1815-0659, Link, MathReview (Renato Álvarez-Nodarse), ZentralBlatt

Referenced by:

§18.38(iii).

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

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

Referenced by:

§18.28(xi).

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4: 18.39 Applications in the Physical Sciences

… ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ►Analogous to (18.39.7) the 3D Schrödinger operator is …where ${\mathrm{L}}^2$ is the (squared) angular momentum operator (14.30.12). … ►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). … ►The radial operator (18.39.28) …

5: Preface

… ► Stegun, editors); and to disseminate essentially the same information from a public website operated by NIST. … ►Dunkl, J. …

6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Bounded and Unbounded Linear Operators

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Self-Adjoint Operators

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Spectrum of an Operator

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7: 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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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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8: 18.36 Miscellaneous Polynomials

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§18.36(i) Jacobi-Type Polynomials

… ►See Liaw et al. (2016, Eqns. 1.1 and 1.2), for the origin of this type characterization. … ►Two representative examples, type I $X_1$-Laguerre, Gómez-Ullate et al. (2010), and type III $X_2$-Hermite, Gómez-Ullate and Milson (2014) EOP’s, are illustrated here. … ►

Type I $X_1$-Laguerre EOP’s

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Type III $X_2$-Hermite EOP’s

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9: Errata

… ►This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order differential operators and eigenfunction expansions. … ►The specific updates to Chapter 1 include the addition of an entirely new subsection §1.18 entitled “Linear Second Order Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order differential operators. The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

33.14.15 $${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}$$

The definite integral, originally written as ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was clarified and rewritten as an orthogonality relation. This follows from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

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A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order $\alpha$ was more precisely identified as the Riemann-Liouville fractional integral operator of order $\alpha$, and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).

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10: 18.28 Askey–Wilson Class

… ► $y$) such that $P_n(z)=p_n(\frac{1}{2}(z+z^{-1}))$ in the Askey–Wilson case, and $P_n(y)=p_n(q^{-y}+c{q}^{y+1})$ in the $q$-Racah case, and both are eigenfunctions of a second order $q$-difference operator similar to (18.27.1). … ►where the operator $L$ is defined by … ►In Tsujimoto et al. (2012) an extension of the Bannai–Ito polynomials occurs as eigenfunctions of a Dunkl type operator. …