Askey–Wilson class orthogonal 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

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2: 18.28 Askey–Wilson Class

§18.28 Askey–Wilson Class

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§18.28(ii) Askey–Wilson Polynomials

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

3: 18.21 Hahn Class: Interrelations

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Figure 18.21.1: Askey scheme. … Magnify

4: 18.26 Wilson Class: Continued

… ►Moreover, if one or more of the new parameters becomes zero, then the polynomial descends to a lower family in the Askey scheme.

5: 18.27 $q$-Hahn Class

§18.27 $q$-Hahn Class

… ►The $q$-hypergeometric OP’s comprise the $q$-Hahn class OP’s and the Askey–Wilson class OP’s (§18.28). … ►All these systems of OP’s have orthogonality properties of the form … ►

§18.27(ii) $q$-Hahn Polynomials

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Discrete $q$-Hermite II

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6: 18.22 Hahn Class: Recurrence Relations and Differences

§18.22 Hahn Class: Recurrence Relations and Differences

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§18.22(i) Recurrence Relations in $n$

… ►These polynomials satisfy (18.22.2) with $p_n(x)$, $A_n$, and $C_n$ as in Table 18.22.1. … ►

§18.22(ii) Difference Equations in $x$

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§18.22(iii) $x$-Differences

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7: 18.1 Notation

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Hahn Class OP’s

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Wilson Class OP’s

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Wilson: $W_n(x;a,b,c,d)$.

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Askey–Wilson Class OP’s

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Askey–Wilson: $p_n(x;a,b,c,d|q)$.

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8: Bibliography M

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Symmetric Functions and Orthogonal Polynomials. University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0770-6, MathReview (Angèle M. Hamel), ZentralBlatt

Referenced by:

§17.14.

Encodings:

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Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).

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External Links:

ISSN 1286-4889, FullText, MathReview, ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

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Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-82472-9, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

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Associated Wilson polynomials. Constr. Approx. 7 (4), pp. 521–534.

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External Links:

ISSN 0176-4276, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey)

Referenced by:

§18.27(v).

Encodings:

BibTeX

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9: Bibliography C

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Work Group of Computational Mathematics, University of Kassel, Germany.

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Notes:

Computer Algebra and Orthogonal Polynomials: a Web service using Maple for online generation of formulas and graphs of orthogonal polynomials belonging to the Askey scheme. For further information see Swarttouw (1997) and Koepf (1999).

External Links:

CAOP

Referenced by:

1st item.

Encodings:

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Asymptotics of the largest zeros of some orthogonal polynomials. J. Phys. A 31 (25), pp. 5525–5544.

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External Links:

ISSN 0305-4470, Document, MathReview (Tim H. Baker), ZentralBlatt

Referenced by:

§18.26(v), §18.29.

Encodings:

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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:

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An Introduction to Orthogonal Polynomials. Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York.

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External Links:

ISBN 0-677-04150-0, MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.2(iii), §18.2(iv), §18.20(i), §18.30, §18.30, Ch.18.

Encodings:

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Extremal measures for a system of orthogonal polynomials. Constr. Approx. 9, pp. 111–119.

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External Links:

ISSN 0176-4276, Document, MathReview (Mizan Rahman), ZentralBlatt

Referenced by:

§18.28(iv).

Encodings:

BibTeX

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