relation to generalized hypergeometric functions

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21: 16.25 Methods of Computation

§16.25 Methods of Computation

►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …Instead a boundary-value problem needs to be formulated and solved. …

22: 13.18 Relations to Other Functions

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§13.18(ii) Incomplete Gamma Functions

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§13.18(iv) Parabolic Cylinder Functions

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§13.18(v) Orthogonal Polynomials

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Hermite Polynomials

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Laguerre Polynomials

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23: 35.9 Applications

§35.9 Applications

►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument

{{}_{p}F_{q}}

, with

p\leq 2

and

q\leq 1

. … ►For other statistical applications of

{{}_{p}F_{q}}

functions of matrix argument see Perlman and Olkin (1980), Groeneboom and Truax (2000), Bhaumik and Sarkar (2002), Richards (2004) (monotonicity of power functions of multivariate statistical test criteria), Bingham et al. (1992) (Procrustes analysis), and Phillips (1986) (exact distributions of statistical test criteria). These references all use results related to the integral formulas (35.4.7) and (35.5.8). … ►In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. …

24: 15.17 Mathematical Applications

… ►This topic is treated in §§15.10 and 15.11. … ► … ►For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ► …

25: 14.3 Definitions and Hypergeometric Representations

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14.3.14

w_{2}(\nu,\mu,x)=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1% \right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)}x\left(1% -x^{2}\right)^{-\mu/2}\mathbf{F}\left(\tfrac{1}{2}-\tfrac{1}{2}\nu-\tfrac{1}{2% }\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1;\tfrac{3}{2};x^{2}\right).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $w_{2}$ : function $w_{1}$ ,
Referenced by:: §10.59, §14.3(i), §14.5(i)
Permalink:: http://dlmf.nist.gov/14.3.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §14.3(iii), §14.3 and Ch.14

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26: 15.16 Products

§15.16 Products

… ►where

A_{0}=1

and

A_{s}

s=1,2,\dots

, are defined by the generating function … ►

Generalized Legendre’s Relation

►

15.16.5

F\left({\frac{1}{2}+\lambda,-\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F% \left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)+F\left({% \frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({-\frac% {1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)-F\left({\frac{1}{2}+% \lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({\frac{1}{2}-% \lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)=\frac{\Gamma\left(1+\lambda% +\mu\right)\Gamma\left(1+\nu+\mu\right)}{\Gamma\left(\lambda+\mu+\nu+\frac{3}{% 2}\right)\Gamma\left(\frac{1}{2}+\nu\right)},

|\operatorname{ph}z|<\pi

|\operatorname{ph}\left(1-z\right)|<\pi

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/15.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16, §15.16 and Ch.15

►For further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).

27: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

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§35.6(i) Definitions

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Laguerre Form

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§35.6(ii) Properties

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§35.6(iii) Relations to Bessel Functions of Matrix Argument

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28: 17.4 Basic Hypergeometric Functions

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17.4.2

\lim_{q\to 1-}{{}_{r+1}\phi_{s}}\left({q^{a_{0}},q^{a_{1}},\dots,q^{a_{r}}% \atop q^{b_{1}},\dots,q^{b_{s}}};q,(q-1)^{s-r}z\right)={{}_{r+1}F_{s}}\left({a% _{0},a_{1},\dots,a_{r}\atop b_{1},\dots,b_{s}};z\right).

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $r$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Source:: Koekoek et al. (2010, p. 15)
Referenced by:: Erratum (V1.1.11) for Equation (17.4.2)
Permalink:: http://dlmf.nist.gov/17.4.E2
Encodings:: pMML, png, TeX
Generalization (effective with 1.1.11):: This limit relation which was previously accurate for ${{}_{r+1}\phi_{r}}$ was updated to be accurate for the more general ${{}_{r+1}\phi_{s}}$ .
See also:: Annotations for §17.4(i), §17.4 and Ch.17

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29: 18.27 $q$ -Hahn Class

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§18.27(ii) $q$ -Hahn Polynomials

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§18.27(iii) Big $q$ -Jacobi Polynomials

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§18.27(iv) Little $q$ -Jacobi Polynomials

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§18.27(v) $q$ -Laguerre Polynomials

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Discrete $q$ -Hermite II

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30: 8.27 Approximations

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§8.27(i) Incomplete Gamma Functions

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•

DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}e^{x^{2}/2}\int_{x}^{\infty}e^{-t^{2}/2}t^{p}\,\mathrm{d}t$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$ . This takes the form $F(p,x)=4x/h(p,x)$ , approximately, where $h(p,x)=3(x^{2}-p)+\sqrt{(x^{2}-p)^{2}+8(x^{2}+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to\infty$ .

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•

Luke (1969b, p. 186) gives hypergeometric polynomial representations that converge uniformly on compact subsets of the $z$ -plane that exclude $z=0$ and are valid for $\left|\operatorname{ph}z\right|<\pi$ .

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§8.27(ii) Generalized Exponential Integral

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•

Luke (1975, p. 103) gives Chebyshev-series expansions for $E_{1}\left(x\right)$ and related functions for $x\geq 5$ .

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