hypergeometric representations

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21: 18.28 Askey–Wilson Class

… ► ►In the remainder of this section the Askey–Wilson class OP’s are defined by their $q$-hypergeometric representations, followed by their orthogonal properties. … ►Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).

22: 35.10 Methods of Computation

§35.10 Methods of Computation

… ►Other methods include numerical quadrature applied to double and multiple integral representations. See Yan (1992) for the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $𝐎(m)$ applied to a generalization of the integral (35.5.8). …

23: 16.7 Relations to Other Functions

… ►Further representations of special functions in terms of ${}_p{}^{}F_q^{}$ functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of ${}_{q+1}{}^{}F_q^{}$ functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).

24: 18.27 $q$-Hahn Class

… ►They are defined by their $q$-hypergeometric representations, followed by their orthogonality properties. … ►

18.27.13 $$p_n(x)=p_n(x;a,b;q)={}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},ab{q}^{n+1}}{aq};q,qx)={(-b)}^{-n}{q}^{-n(n+1)/2}\frac{{(qb;q)}_n}{{(qa;q)}_n}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},ab{q}^{n+1},qbx}{qb,0};q,q).$$

ⓘ

Defines:

$p_n(x;a,b;q)$: little $q$-Jacobi polynomial

Symbols:

${(a;q)}_n$: $q$-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, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.28.27), (18.28.28), Erratum (V1.2.0) for Equation (18.27.13)

Permalink:

http://dlmf.nist.gov/18.27.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

The ${}_3{}^{}\varphi _2^{}$ representation was added.

See also:

Annotations for §18.27(iv), §18.27 and Ch.18

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25: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

Integral Representation

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26: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

… ►

§17.6(v) Integral Representations

… ►For continued-fraction representations of the ${}_2{}^{}\varphi _1^{}$ function, see Cuyt et al. (2008, pp. 395–399).

27: 13.25 Products

… ►For integral representations, integrals, and series containing products of $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ see Erdélyi et al. (1953a, §6.15.3).

28: 18.26 Wilson Class: Continued

… ►

§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

…

29: 13.12 Products

… ►For integral representations, integrals, and series containing products of $M(a,b,z)$ and $U(a,b,z)$ see Erdélyi et al. (1953a, §6.15.3).

30: 16.24 Physical Applications

§16.24 Physical Applications

►

§16.24(i) Random Walks

►Generalized hypergeometric functions and Appell functions appear in the evaluation of the so-called Watson integrals which characterize the simplest possible lattice walks. … ►

§16.24(iii) $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ Symbols

►The $\mathit{3}j$ symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. …