hypergeometric representations

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31: 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. …

32: 15.19 Methods of Computation

… ►The representation (15.6.1) can be used to compute the hypergeometric function in the sector

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

. …

33: 16.8 Differential Equations

… ►In this reference it is also explained that in general when

q>1

no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near

z=1

. …

34: 16.17 Definition

… ►Then the Meijer $G$ -function is defined via the Mellin–Barnes integral representation: … ►

…

Figure 16.17.1: s-plane. Path

L

for the integral representation (16.17.1) of the Meijer

G

-function.

… ►Then ►

16.17.2

{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\sum_% {k=1}^{m}A_{p,q,k}^{m,n}(z){{}_{p}F_{q-1}}\left({1+b_{k}-a_{1},\dots,1+b_{k}-a% _{p}\atop 1+b_{k}-b_{1},\ldots*\dots,1+b_{k}-b_{q}};(-1)^{p-m-n}z\right),

ⓘ

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, ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $A_{p,q}^{m,n}(z)$ : coefficient
Permalink:: http://dlmf.nist.gov/16.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.17 and Ch.16

…

35: 13.16 Integral Representations

§13.16 Integral Representations

►

§13.16(i) Integrals Along the Real Line

… ► ►

§13.16(ii) Contour Integrals

►For contour integral representations combine (13.14.2) and (13.14.3) with §13.4(ii). …

36: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

… ►For continued-fraction representations of a ratio of

{{}_{3}\phi_{2}}

functions, see Cuyt et al. (2008, pp. 399–400). …

37: 12.18 Methods of Computation

… ►Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs. These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …

38: Donald St. P. Richards

… ►He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T. …

39: 13.4 Integral Representations

§13.4 Integral Representations

… ►

§13.4(ii) Contour Integrals

… ► … ►

§13.4(iii) Mellin–Barnes Integrals

… ►

40: 15.4 Special Cases

… ►

§15.4(i) Elementary Functions

… ►For an extensive list of elementary representations see Prudnikov et al. (1990, pp. 468–488). … ►

Chu–Vandermonde Identity

… ►

§15.4(iii) Other Arguments

…