§16.18Special Cases

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Keywords:: Meijer $G$ -function,generalized hypergeometric functions,relation to generalized hypergeometric function,relations to other functions,special cases
Notes:: SeeLuke (1969a, Chapter 5).
Permalink:: http://dlmf.nist.gov/16.18
See also:: Annotations forCh.16

The ${{}_{1}F_{1}}$ and ${{}_{2}F_{1}}$ functions introduced in Chapters13 and15, as well as the more general ${{}_{p}F_{q}}$ functions introduced in the present chapter, are all special cases of theMeijer $G$ -function. This is a consequence of the following relations:

16.18.1		${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\left({%\textstyle\ifrac{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_%{k=1}^{p}\Gamma\left(a_{k}\right)}}\right){G^{1,p}_{p,q+1}}\left(-z;{1-a_{1},%\dots,1-a_{p}\atop 0,1-b_{1},\dots,1-b_{q}}\right)=\left({\textstyle\ifrac{%\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_{k=1}^{p}\Gamma%\left(a_{k}\right)}}\right){G^{p,1}_{q+1,p}}\left(-\frac{1}{z};{1,b_{1},\dots,%b_{q}\atop a_{1},\dots,a_{p}}\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_%{q}};\NVar{z}\right)$ 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 and $b,b_{1},\ldots,b_{q}$ : real or complex parameters Permalink: http://dlmf.nist.gov/16.18.E1 Encodings: TeX,pMML,png See also: Annotations for§16.18 andCh.16

As a corollary, special cases of the ${{}_{1}F_{1}}$ and ${{}_{2}F_{1}}$ functions, including Airy functions, Bessel functions, parabolic cylinderfunctions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, areall special cases of the Meijer $G$ -function. Representations of specialfunctions in terms of the Meijer $G$ -function are given inErdélyi et al. (1953a, §5.6),Luke (1969a, §§6.4–6.5), andMathai (1993, §3.10).

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§16.18Special Cases