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31: 18.33 Polynomials Orthogonal on the Unit Circle

§18.33 Polynomials Orthogonal on the Unit Circle

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§18.33(i) Definition

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§18.33(iii) Connection with OP’s on the Line

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§18.33(v) Biorthogonal Polynomials on the Unit Circle

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Recurrence Relations

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32: 18.13 Continued Fractions

§18.13 Continued Fractions

… ►See also Cuyt et al. (2008, pp. 91–99).

33: 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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34: 18.25 Wilson Class: Definitions

§18.25 Wilson Class: Definitions

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Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

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OP	$p_n(x)$	$x=\lambda (y)$	Orthogonality range for $y$	Constraints
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§18.25(ii) Weights and Standardizations: Continuous Cases

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18.25.15 $$h_n=\frac{n!(N-n)!{\left(\gamma +\delta +2\right)}_N}{N!{\left(\gamma +1\right)}_n{\left(\delta +1\right)}_{N-n}}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $N$: positive integer, $\delta$: arbitrary small positive constant, $n$: nonnegative integer and $h_n$

Referenced by:

§18.25(iii)

Permalink:

http://dlmf.nist.gov/18.25.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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Table 18.25.2: Wilson class OP’s: leading coefficients.

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$p_n(x)$	$k_n$
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35: 1 Algebraic and Analytic Methods

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36: 14 Legendre and Related Functions

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37: Peter A. Clarkson

… ►He is a member of the editorial boards of nine international journals and has served as Chair, Vice-Chair, and Secretary of the SIAM Activity Group on Orthogonal Polynomials and Special Functions. …

38: 18.30 Associated OP’s

§18.30 Associated OP’s

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§18.30(vi) Corecursive Orthogonal Polynomials

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Numerator and Denominator Polynomials

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§18.30(vii) Corecursive and Associated Monic Orthogonal Polynomials

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39: 18.26 Wilson Class: Continued

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§18.26(ii) Limit Relations

… ►See also Figure 18.21.1. ►

§18.26(iii) Difference Relations

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§18.26(iv) Generating Functions

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§18.26(v) Asymptotic Approximations

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40: 18.20 Hahn Class: Explicit Representations

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§18.20(i) Rodrigues Formulas

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§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

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