affine Weyl groups
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1: Bibliography N
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Numerical Algorithms Group, Ltd..
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Askey-Wilson polynomials: an affine Hecke algebra approach.
In Laredo Lectures on Orthogonal Polynomials and Special
Functions,
Adv. Theory Spec. Funct. Orthogonal Polynomials, pp. 111–144.
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Affine Weyl groups, discrete dynamical systems and Painlevé equations.
Comm. Math. Phys. 199 (2), pp. 281–295.
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2: 32.7 Bäcklund Transformations
3: Bibliography K
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The group
, separation of variables and the hydrogen atom.
SIAM J. Appl. Math. 30 (4), pp. 630–664.
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Quantum Groups.
Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York.
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Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups.
In Special Functions: Group Theoretical Aspects and Applications,
pp. 1–85.
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Compact quantum groups and -special functions.
In Representations of Lie Groups and Quantum Groups,
Pitman Res. Notes Math. Ser., Vol. 311, pp. 46–128.
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The relationship between Zhedanov’s algebra and the double affine Hecke algebra in the rank one case.
SIGMA 3, pp. Paper 063, 15 pp..
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4: 18.38 Mathematical Applications
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Group Representations
►For group-theoretic interpretations of OP’s see Vilenkin and Klimyk (1991, 1992, 1993). … ►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. …In the -case this algebraic structure is called the double affine Hecke algebra (DAHA), introduced by Cherednik. …5: 13.27 Mathematical Applications
§13.27 Mathematical Applications
►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. The elements of this group are of the form …The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. … …6: 17.17 Physical Applications
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►See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys.
►Quantum groups also apply -series extensively.
Quantum groups are really not groups at all but certain Hopf algebras.
They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics.
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7: 15.17 Mathematical Applications
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§15.17(iii) Group Representations
… ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►§15.17(v) Monodromy Groups
… ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …8: Bibliography M
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Affine root systems and Dedekind’s -function.
Invent. Math. 15 (2), pp. 91–143.
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Affine Hecke Algebras and Orthogonal Polynomials.
Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge.
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The Theory of Groups.
Clarendon Press, Oxford.
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Crossing symmetric expansions of physical scattering amplitudes: The
group and Lamé functions.
J. Mathematical Phys. 12, pp. 281–293.
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Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.
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9: 20.12 Mathematical Applications
10: Tom H. Koornwinder
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►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC.
Books for which he has been editor or coeditor include Special Functions: Group Theoretical Aspects and Applications (with R.
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►Koornwinder has been active as an officer in the SIAM Activity Group on Special Functions and Orthogonal Polynomials.
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