Wilson polynomials

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1: 18.26 Wilson Class: Continued

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18.26.5

\lim_{d\to\infty}\frac{W_{n}\left(x;a,b,c,d\right)}{{\left(a+d\right)_{n}}}=S_% {n}\left(x;a,b,c\right).

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : continuous dual Hahn polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.26.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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18.26.14

\ifrac{\delta_{y}\left(W_{n}\left(y^{2};a,b,c,d\right)\right)}{\delta_{y}(y^{2% })}=-n(n+a+b+c+d-1)\*W_{n-1}\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac% {1}{2},d+\tfrac{1}{2}\right).

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Symbols:: $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $\delta_{\NVar{x}}$ : central difference, $y$ : real variable and $n$ : nonnegative integer
Referenced by:: §16.4(iii), §18.26(iii)
Permalink:: http://dlmf.nist.gov/18.26.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iii), §18.26 and Ch.18

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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

W_{n}\left(x;a,b,c,d\right)

, continuous dual Hahn polynomials

S_{n}\left(x;a,b,c\right)

, Racah polynomials

R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)

, and dual Hahn polynomials

R_{n}\left(x;\gamma,\delta,N\right)

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Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

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OP	$p_{n}(x)$	$x=\lambda(y)$	Orthogonality range for $y$	Constraints
Wilson	$W_{n}\left(x;a,b,c,d\right)$	$y^{2}$	$(0,\infty)$	$\Re(a,b,c,d)>0$ ; nonreal parameters in conjugate pairs
…

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18.25.3

p_{n}(x)=W_{n}\left(x;a_{1},a_{2},a_{3},a_{4}\right),

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Symbols:: $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.25(ii)
Permalink:: http://dlmf.nist.gov/18.25.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

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Table 18.25.2: Wilson class OP’s: leading coefficients.

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$p_{n}(x)$	$k_{n}$
$W_{n}\left(x;a,b,c,d\right)$	$(-1)^{n}{\left(n+a+b+c+d-1\right)_{n}}$
…

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3: 18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for $q$ -Hahn and Askey–Wilson Classes

►Ismail (1986) gives asymptotic expansions as

n\to\infty

, with

x

and other parameters fixed, for continuous

q

-ultraspherical, big and little

q

-Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson

p_{n}\left(\cos\theta;a,b,c,d\,|\,q\right)

the leading term is given by …

4: 18.28 Askey–Wilson Class

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§18.28(ii) Askey–Wilson Polynomials

… ►The polynomials

p_{n}\left(x;a,b,c,d\,|\,q\right)

are symmetric in the parameters

a, b, c, d

. … ►

$q$ -Difference Equation

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Recurrence Relation

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Duality

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5: 18.1 Notation

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\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.

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Wilson Class OP’s

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Wilson: $W_{n}\left(x;a,b,c,d\right)$ .

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Askey–Wilson: $p_{n}\left(x;a,b,c,d\,|\,q\right)$ .

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6: Tom H. Koornwinder

… ►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. …

7: Richard A. Askey

… ► Wilson), introduced the Askey-Wilson polynomials. …

8: 18.38 Mathematical Applications

… ► … ►If we consider this abstract algebra with additional relation (18.38.9) and with dependence on

a, b, c, d

according to (18.38.7) then it is isomorphic with the algebra generated by

K_{0}=L

given by (18.28.6_2),

(K_{1}f)(z)=(z+z^{-1})f(z)

and

K_{2}

given by (18.38.4), and

K_{0},K_{1},K_{2}

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). … ►The Dunkl type operator is a

q

-difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial

R_{n}(z;a,b,c,d\,|\,q)

and the ‘anti-symmetric’ Laurent polynomial

z^{-1}(1-az)(1-bz)R_{n-1}(z;qa,qb,c,d\,|\,q)

, where

R_{n}(z)

is given in (18.28.1_5). … ►Dunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and

q

-Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7). …

9: 18.21 Hahn Class: Interrelations

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§18.21(i) Dualities

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See accompanying text — Figure 18.21.1: Askey scheme. …It increases by one for each row ascended in the scheme, culminating with four free real parameters for the Wilson and Racah polynomials. … Magnify

10: Bibliography K

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T. H. Koornwinder and F. Bouzeffour (2011) Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials. Appl. Anal. 90 (3-4), pp. 731–746.

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External Links:: ISSN 0003-6811, Link, MathReview (Masatoshi Iida), ZentralBlatt
Referenced by:: §18.38(iii), §18.38(iii).
Encodings:: BibTeX

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T. H. Koornwinder (1992) Askey-Wilson Polynomials for Root Systems of Type

B C

. 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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External Links:: FullText, MathReview (Eric M. Opdam), ZentralBlatt
Referenced by:: §18.37(iii).
Encodings:: BibTeX

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T. H. Koornwinder (1993) Askey-Wilson polynomials as zonal spherical functions on the

{\rm SU}(2)

quantum group. SIAM J. Math. Anal. 24 (3), pp. 795–813.

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External Links:: ISSN 0036-1410, Document, Link, MathReview Entry
Referenced by:: (18.28.27).
Encodings:: BibTeX

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T. H. Koornwinder (2007b) The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 207 (2), pp. 214–226.

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External Links:: ISSN 0377-0427, Document, Link, MathReview Entry
Referenced by:: (18.9.18_5).
Encodings:: BibTeX

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T. H. Koornwinder (2012) Askey-Wilson polynomial. Scholarpedia 7 (7), pp. 7761.

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Notes:: Electronic only
External Links:: Document, FullText
Referenced by:: §18.28(i), §18.28(i).
Encodings:: BibTeX

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