Wilson polynomials
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1: 18.26 Wilson Class: Continued
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18.26.5
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18.26.14
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§18.26(iv) Generating Functions
… ►For asymptotic expansions of Wilson polynomials of large degree see Wilson (1991), and for asymptotic approximations to their largest zeros see Chen and Ismail (1998). ►Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. …2: 18.25 Wilson Class: Definitions
§18.25 Wilson Class: Definitions
… ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials , continuous dual Hahn polynomials , Racah polynomials , and dual Hahn polynomials . ►OP | Orthogonality range for | Constraints | ||
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Wilson | ; nonreal parameters in conjugate pairs | |||
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18.25.3
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3: 18.29 Asymptotic Approximations for -Hahn and Askey–Wilson Classes
§18.29 Asymptotic Approximations for -Hahn and Askey–Wilson Classes
►Ismail (1986) gives asymptotic expansions as , with and other parameters fixed, for continuous -ultraspherical, big and little -Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson the leading term is given by …4: 18.28 Askey–Wilson Class
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§18.28(ii) Askey–Wilson Polynomials
… ►The polynomials are symmetric in the parameters . … ►-Difference Equation
… ►Recurrence Relation
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…5: 18.1 Notation
6: 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.
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7: Richard A. Askey
8: 18.38 Mathematical Applications
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►If we consider this abstract algebra with additional relation (18.38.9) and with dependence on according to (18.38.7) then it is isomorphic with the algebra generated by given by (18.28.6_2), and given by (18.38.4), and act on the linear span of the Askey–Wilson polynomials (18.28.1).
See Zhedanov (1991), Granovskiĭ et al. (1992, §3), Koornwinder (2007a, §2) and Terwilliger (2011).
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►The Dunkl type operator is a -difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial
and the ‘anti-symmetric’ Laurent polynomial
, where is given in (18.28.1_5).
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►Dunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and -Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7).
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9: 18.21 Hahn Class: Interrelations
10: Bibliography K
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Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials.
Appl. Anal. 90 (3-4), pp. 731–746.
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Askey-Wilson Polynomials for Root Systems of Type
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In Hypergeometric Functions on Domains of Positivity, Jack
Polynomials, and Applications (Tampa, FL, 1991),
Contemp. Math., Vol. 138, pp. 189–204.
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Askey-Wilson polynomials as zonal spherical functions on the quantum group.
SIAM J. Math. Anal. 24 (3), pp. 795–813.
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The structure relation for Askey-Wilson polynomials.
J. Comput. Appl. Math. 207 (2), pp. 214–226.
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Askey-Wilson polynomial.
Scholarpedia 7 (7), pp. 7761.
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