# Askey 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 ${p}_{n}(\mathrm{cos}\theta ;a,b,c,d|q)$ the leading term is given by …##### 2: 18.28 Askey–Wilson Class

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►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.
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###### §18.28(ii) Askey–Wilson Polynomials

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18.28.1
$${p}_{n}(\mathrm{cos}\theta )={p}_{n}(\mathrm{cos}\theta ;a,b,c,d|q)={a}^{-n}\sum _{\mathrm{\ell}=0}^{n}{q}^{\mathrm{\ell}}{(ab{q}^{\mathrm{\ell}},ac{q}^{\mathrm{\ell}},ad{q}^{\mathrm{\ell}};q)}_{n-\mathrm{\ell}}\frac{{({q}^{-n},abcd{q}^{n-1};q)}_{\mathrm{\ell}}}{{(q;q)}_{\mathrm{\ell}}}\prod _{j=0}^{\mathrm{\ell}-1}(1-2a{q}^{j}\mathrm{cos}\theta +{a}^{2}{q}^{2j})={a}^{-n}{(ab,ac,ad;q)}_{n}{}_{4}\varphi _{3}(\genfrac{}{}{0pt}{}{{q}^{-n},abcd{q}^{n-1},a{\mathrm{e}}^{\mathrm{i}\theta},a{\mathrm{e}}^{-\mathrm{i}\theta}}{ab,ac,ad};q,q).$$

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►For ${\omega}_{y}$ and ${h}_{n}$ see Koekoek et al. (2010, Eq. (14.2.2)).
##### 3: 18.33 Polynomials Orthogonal on the Unit Circle

##### 4: 18.1 Notation

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Askey–Wilson: ${p}_{n}(x;a,b,c,d|q)$.

##### 5: 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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##### 6: 18.21 Hahn Class: Interrelations

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##### 7: Richard A. Askey

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► 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.
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►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.
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►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.
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##### 8: Bibliography

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Recurrence relations, continued fractions, and orthogonal polynomials.
Mem. Amer. Math. Soc. 49 (300), pp. iv+108.
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An integral for Jacobi polynomials.
Simon Stevin 46, pp. 165–169.
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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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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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Continuous Hahn polynomials.
J. Phys. A 18 (16), pp. L1017–L1019.
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##### 9: 18.30 Associated OP’s

##### 10: Bibliography K

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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.
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Two-variable Analogues of the Classical Orthogonal Polynomials.
In Theory and Application of Special Functions, R. A. Askey (Ed.),
pp. 435–495.
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Askey-Wilson Polynomials for Root Systems of Type $BC$
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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 polynomial.
Scholarpedia 7 (7), pp. 7761.
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