Askey–Wilson polynomials

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1: 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 $n\to \mathrm{\infty }$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson $p_n(\cos \theta ;a,b,c,d|q)$ the leading term is given by …

2: 18.28 Askey–Wilson Class

§18.28 Askey–Wilson Class

… ►Both the Askey–Wilson polynomials and the $q$-Racah polynomials can best be described as functions of $z$ (resp. … ►

§18.28(ii) Askey–Wilson Polynomials

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18.28.1 $$p_n(\cos \theta )=p_n(\cos \theta ;a,b,c,d|q)=a^{-n}\sum _{\mathrm{\ell }=0}^n{q}^{\mathrm{\ell }}{(ab{q}^{\mathrm{\ell }},ac{q}^{\mathrm{\ell }},ad{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}\frac{{(q^{-n},abcd{q}^{n-1};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^j\cos \theta +a^2{q}^{2j})=a^{-n}{(ab,ac,ad;q)}_n{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},abcd{q}^{n-1},a{\mathrm{e}}^{\mathrm{i}\theta },a{\mathrm{e}}^{-\mathrm{i}\theta }}{ab,ac,ad};q,q).$$

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Defines:

$p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial

Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(ii), §18.28 and Ch.18

… ►For ${\omega }_y$ and $h_n$ see Koekoek et al. (2010, Eq. (14.2.2)).

3: 18.1 Notation

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Askey–Wilson: $p_n(x;a,b,c,d|q)$.

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4: 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. …

5: Richard A. Askey

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

6: 18.21 Hahn Class: Interrelations

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Figure 18.21.1: Askey scheme. … Magnify

7: 18.30 Associated OP’s

… ►For associated Askey–Wilson polynomials see Rahman (2001).

8: Bibliography K

… ►

Askey-Wilson Polynomials for Root Systems of Type $BC$ . 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

… ►

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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9: 18.37 Classical OP’s in Two or More Variables

… ►In one variable they are essentially ultraspherical, Jacobi, continuous $q$-ultraspherical, or Askey–Wilson polynomials. …

10: Bibliography C

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On the zeros of the Askey-Wilson polynomials, with applications to coding theory. SIAM J. Math. Anal. 18 (1), pp. 191–207.

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External Links:

ISSN 0036-1410, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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