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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Differential-difference operators associated to reflection groups.
Trans. Amer. Math. Soc. 311 (1), pp. 167–183.
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A Laguerre polynomial orthogonality and the hydrogen atom.
Anal. Appl. (Singap.) 1 (2), pp. 177–188.
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Special Functions.
World Scientific Publishing Co., Inc., River Edge, NJ.
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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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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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Numerical algorithms for uniform Airy-type asymptotic expansions.
Numer. Algorithms 15 (2), pp. 207–225.
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The universal Askey-Wilson algebra and DAHA of type
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SIGMA 9, pp. Paper 047, 40 pp..
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Dunkl shift operators and Bannai-Ito polynomials.
Adv. Math. 229 (4), pp. 2123–2158.
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4: 18.39 Applications in the Physical Sciences
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►The fundamental quantum Schrödinger operator, also called the Hamiltonian, , is a second order differential operator of the form
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►Analogous to (18.39.7) the 3D Schrödinger operator is
…where is the (squared) angular momentum operator (14.30.12).
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►noting that the 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).
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►The radial operator (18.39.28)
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5: Preface
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► Stegun, editors); and to disseminate essentially the same information from a public website operated by NIST.
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►Dunkl, J.
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6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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Bounded and Unbounded Linear Operators
… ► … ► … ►Self-Adjoint Operators
… ►Spectrum of an Operator
…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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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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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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On the foundations of combinatorial theory. VIII. Finite operator calculus.
J. Math. Anal. Appl. 42, pp. 684–760.
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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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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 -Laguerre, Gómez-Ullate et al. (2010), and type III -Hermite, Gómez-Ullate and Milson (2014) EOP’s, are illustrated here. … ►Type I -Laguerre EOP’s
… ►Type III -Hermite EOP’s
…9: Errata
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►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.
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►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.
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Equation (33.14.15)
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Subsections 1.15(vi), 1.15(vii), 2.6(iii)
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33.14.15
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 was more precisely identified as the Riemann-Liouville fractional integral operator of order , 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).
10: 18.28 Askey–Wilson Class
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) such that in the Askey–Wilson case, and in the -Racah case, and both are eigenfunctions of a second order -difference operator similar to (18.27.1).
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►where the operator
is defined by
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►In Tsujimoto et al. (2012) an extension of the Bannai–Ito polynomials occurs as eigenfunctions of a Dunkl type operator.
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