Askey%E2%80%93Wilson%20polynomials

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11: 18.28 Askey–Wilson Class

§18.28 Askey–Wilson Class

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§18.28(ii) Askey–Wilson Polynomials

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Recurrence Relation

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Duality

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From Askey–Wilson to Wilson

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12: 18 Orthogonal Polynomials

Chapter 18 Orthogonal Polynomials

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13: 18.38 Mathematical Applications

… ►The Askey–Gasper inequality … ►See Zhedanov (1991), Granovskiĭ et al. (1992, §3), Koornwinder (2007a, §2) and Terwilliger (2011). Similar algebras can be associated with all families of OP’s in the $q$-Askey scheme and the Askey scheme. … ►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). …

14: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

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§18.21(i) Dualities

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§18.21(ii) Limit Relations and Special Cases

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Hahn $\to$ Jacobi

… ►A graphical representation of limits in §§18.7(iii), 18.21(ii), and 18.26(ii) is provided by the Askey scheme depicted in Figure 18.21.1. …

15: Bibliography H

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Une $q$-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.

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External Links:

ISSN 0036-1410, Document, MathReview (R. A. Askey), ZentralBlatt

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§17.13.

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:

MathReview (L. Berg)

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§7.14(iii).

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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

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Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

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Asymptotik bei Jacobi-Polynomen und Jacobi-Funktionen. Math. Z. 171 (3), pp. 201–226 (German).

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ISSN 0025-5874, Document, MathReview (R. A. Askey), ZentralBlatt

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§18.15(i).

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16: Bibliography I

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On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt

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§18.36(ii).

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Two families of associated Wilson polynomials. Canad. J. Math. 42 (4), pp. 659–695.

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ISSN 0008-414X, MathReview (Mizan Rahman), ZentralBlatt

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§18.26(iv).

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$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt

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§18.28(vii).

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Asymptotics of the Askey-Wilson and $q$-Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.

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ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt

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§18.29.

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Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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Notes:

With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.

External Links:

ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt

Referenced by:

§18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

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17: Bibliography D

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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

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ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

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§24.12(iii).

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Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

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Notes:

With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)

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ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt

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§21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.

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Error analysis in a uniform asymptotic expansion for the generalised exponential integral. J. Comput. Appl. Math. 80 (1), pp. 127–161.

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ISSN 0377-0427, Document, MathReview (Eberhard Wagner), ZentralBlatt

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§2.11(iii), §8.20(ii).

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Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

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ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt

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§18.24.

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Nicholson-type Integrals for Products of Gegenbauer Functions and Related Topics. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), R. A. Askey (Ed.), pp. 353–374. Math. Res. Center, Univ. Wisconsin, Publ. No. 35.

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External Links:

MathReview (K. M. Saksena), ZentralBlatt

Referenced by:

(12.12.4), §12.12, §18.17(iii), §18.17(iii).

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18: 5 Gamma Function

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19: 16 Generalized Hypergeometric Functions & Meijer G-Function

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20: Bibliography T

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Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.

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External Links:

ISBN 0-940600-05-6, FullText, MathReview (A. W. Davis), ZentralBlatt

Referenced by:

§35.4(i), §35.9.

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LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:

Single-precision Fortran.

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ISSN 0010-4655, Document, Code

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9th item.

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The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis. J. Comput. Appl. Math. 133 (1-2), pp. 623–633.

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External Links:

ISSN 0377-0427, Document, MathReview (P. N. Rathie), ZentralBlatt

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§18.24.

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The universal Askey-Wilson algebra. SIGMA 7, pp. Paper 069, 24 pp..

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External Links:

ISSN 1815-0659, Link, MathReview Entry, ZentralBlatt

Referenced by:

§18.38(iii).

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The universal Askey-Wilson algebra and DAHA of type $(C_1^{\vee },C_1)$ . SIGMA 9, pp. Paper 047, 40 pp..

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External Links:

ISSN 1815-0659, Link, MathReview (Renato Álvarez-Nodarse), ZentralBlatt

Referenced by:

§18.38(iii).

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