Askey%E2%80%93Wilson%20polynomials

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1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

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31.5.2 $${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

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Defines:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.5.E2

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See also:

Annotations for §31.5 and Ch.31

►is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. These solutions are the Heun polynomials. …

2: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

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Normalization

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Orthogonal Invariance

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Summation

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Mean-Value

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3: Richard A. Askey

Profile
Richard A. Askey

… ►Richard A. Askey (b. … ►Askey received his Ph. … Wilson), introduced the Askey-Wilson polynomials. …Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. …

4: 24.1 Special Notation

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Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►The notations $E_n$, $E_n\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

5: 18.3 Definitions

§18.3 Definitions

… ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►

Bessel polynomials

►Bessel polynomials are often included among the classical OP’s. …

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. … Askey and W. … ►Koornwinder has been active as an officer in the SIAM Activity Group on Special Functions and Orthogonal Polynomials. Currently he is on the editorial board for Constructive Approximation, and is editor for the volume on Multivariable Special Functions in the ongoing Askey–Bateman book project. … ►

18 Orthogonal Polynomials

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7: Bibliography

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Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.

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External Links:

FullText, MathReview (F. Schipp), ZentralBlatt

Referenced by:

§16.23(iii), (18.14.25), (18.14.26), (18.14.27), (18.38.3), Profile Richard A. Askey.

Encodings:

BibTeX

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An integral for Jacobi polynomials. Simon Stevin 46, pp. 165–169.

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External Links:

MathReview (H. M. Srivastava), ZentralBlatt

Referenced by:

§18.17(viii).

Encodings:

BibTeX

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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

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External Links:

ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.

Encodings:

BibTeX

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Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA.

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External Links:

ISBN 0-89871-018-9, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(ii), §18.18(vii), Profile Richard A. Askey.

Encodings:

BibTeX

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Continuous Hahn polynomials. J. Phys. A 18 (16), pp. L1017–L1019.

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External Links:

ISSN 0305-4470, FullText, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.19, §18.20(ii), §18.21(ii).

Encodings:

BibTeX

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8: 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 … ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the $q$-Laguerre and continuous $q^{-1}$-Hermite polynomials see Chen and Ismail (1998).

9: 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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External Links:

ISSN 0003-6811, Link, MathReview (Masatoshi Iida), ZentralBlatt

Referenced by:

§18.38(iii), §18.38(iii).

Encodings:

BibTeX

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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

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Askey-Wilson polynomials as zonal spherical functions on the $\mathrm{SU}(2)$ 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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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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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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10: 1 Algebraic and Analytic Methods

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