Askey polynomials

SearchAdvancedHelp

(0.003 seconds)

1—10 of 38 matching pages

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

… ►The Askey–Wilson class OP’s comprise the four-parameter families of Askey–Wilson polynomials and of $q$-Racah polynomials, and cases of these families obtained by specialization of parameters. …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

►

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

ⓘ

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: 18.33 Polynomials Orthogonal on the Unit Circle

… ►

Szegő–Askey

… ►

Askey

… ► …

4: 18.1 Notation

… ►

Askey–Wilson: $p_n(x;a,b,c,d|q)$.

…

5: 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. …

6: 18.21 Hahn Class: Interrelations

… ►

►►

Figure 18.21.1: Askey scheme. … Magnify

7: Richard A. Askey

… ► Wilson), introduced the Askey-Wilson polynomials. Published in 1985 in the Memoirs of the American Mathematical Society, it also introduced the directed graph of hypergeometric orthogonal polynomials commonly known as the Askey scheme. Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. … ►Additional books for which Askey served as author or editor include Orthogonal Polynomials and Special Functions, published by SIAM in 1975, Theory and application of special functions, published by Academic Press in 1975, Special Functions: Group Theoretical Aspects and Applications (with T. … ►Askey was presented a Lifetime Achievement Award in Recognition and Appreciation for his Outstanding Work and Leadership in the Field of Special Functions at the International Symposium on Orthogonal Polynomials, Special Functions and Applications in Hagenberg, Austria on July 24, 2019. …

8: Bibliography

… ►

Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49 (300), pp. iv+108.

ⓘ

External Links:

ISSN 0065-9266, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§18.28(iv).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

MathReview (H. M. Srivastava)

Referenced by:

§18.17(viii).

Encodings:

BibTeX

►

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

ⓘ

External Links:

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

Referenced by:

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

Encodings:

BibTeX

… ►

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

ⓘ

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

… ►

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

ⓘ

External Links:

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

Referenced by:

§18.19, §18.20(ii), §18.21(ii).

Encodings:

BibTeX

…

9: 18.30 Associated OP’s

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

10: Bibliography K

… ►

The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue. Technical report Technical Report 98-17, Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics.

ⓘ

External Links:

FullText

Referenced by:

R. Koekoek, P. A. Lesky, and R. F. Swarttouw (2010).

Encodings:

BibTeX

… ►

Two-variable Analogues of the Classical Orthogonal Polynomials. In Theory and Application of Special Functions, R. A. Askey (Ed.), pp. 435–495.

ⓘ

External Links:

MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.37(ii).

Encodings:

BibTeX

… ►

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.

ⓘ

External Links:

FullText, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

… ►

Askey-Wilson polynomial. Scholarpedia 7 (7), pp. 7761.

ⓘ

Notes:

Electronic only

External Links:

Document, FullText

Referenced by:

§18.28(i), §18.28(i).

Encodings:

BibTeX

…