16 Generalized Hypergeometric Functions & Meijer G-Function Meijer

G

-Function 16.18 Special Cases 16.20 Integrals and Series

§16.19 Identities

ⓘ

Keywords:: Meijer $G$ -function, identities
Notes:: All of the results in this section are straightforward consequences of the definition (16.17.1).
Permalink:: http://dlmf.nist.gov/16.19
See also:: Annotations for Ch.16

16.19.1	$\displaystyle{G^{m,n}_{p,q}}\left(\frac{1}{z};{a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}}\right)$	$\displaystyle={G^{n,m}_{q,p}}\left(z;{1-b_{1},\dots,1-b_{q}\atop 1-a_{1},\dots% ,1-a_{p}}\right),$
ⓘ Symbols: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ 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 Referenced by: §16.21 Permalink: http://dlmf.nist.gov/16.19.E1 Encodings: TeX, pMML, png See also: Annotations for §16.19 and Ch.16
16.19.2	$\displaystyle z^{\mu}{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}}\right)$	$\displaystyle={G^{m,n}_{p,q}}\left(z;{a_{1}+\mu,\dots,a_{p}+\mu\atop b_{1}+\mu% ,\dots,b_{q}+\mu}\right),$
ⓘ Symbols: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ 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/16.19.E2 Encodings: TeX, pMML, png See also: Annotations for §16.19 and Ch.16
16.19.3	$\displaystyle{G^{m,n+1}_{p+1,q+1}}\left(z;{a_{0},\dots,a_{p}\atop b_{1},\dots,% b_{q},a_{0}}\right)$	$\displaystyle={G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}% }\right),$
ⓘ Symbols: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ 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/16.19.E3 Encodings: TeX, pMML, png See also: Annotations for §16.19 and Ch.16

16.19.4		${G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\frac% {2^{p+1+b_{1}+\dots+b_{q}-m-n-a_{1}-\dots-a_{p}}}{\pi^{m+n-\frac{1}{2}(p+q)}}{% G^{2m,2n}_{2p,2q}}\left(2^{2p-2q}z^{2};{\frac{1}{2}a_{1},\frac{1}{2}a_{1}+% \frac{1}{2},\dots,\frac{1}{2}a_{p},\frac{1}{2}a_{p}+\frac{1}{2}\atop\frac{1}{2% }b_{1},\frac{1}{2}b_{1}+\frac{1}{2},\dots,\frac{1}{2}b_{q},\frac{1}{2}b_{q}+% \frac{1}{2}}\right),$
ⓘ Symbols: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $\pi$ : the ratio of the circumference of a circle to its diameter, $p$ : 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/16.19.E4 Encodings: TeX, pMML, png See also: Annotations for §16.19 and Ch.16

16.19.5		$\vartheta{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}% \right)={G^{m,n}_{p,q}}\left(z;{a_{1}-1,a_{2},\dots,a_{p}\atop b_{1},\dots,b_{% q}}\right)+(a_{1}-1){G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots% ,b_{q}}\right),$
ⓘ Symbols: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ 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 $\vartheta$ : differential operator Permalink: http://dlmf.nist.gov/16.19.E5 Encodings: TeX, pMML, png See also: Annotations for §16.19 and Ch.16

16.19.6		$\int_{0}^{1}t^{-a_{0}}(1-t)^{a_{0}-b_{q+1}-1}{G^{m,n}_{p,q}}\left(zt;{a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}}\right)\,\mathrm{d}t=\Gamma\left(a_{0}-b_{q% +1}\right){G^{m,n+1}_{p+1,q+1}}\left(z;{a_{0},\dots,a_{p}\atop b_{1},\dots,b_{% q+1}}\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p$ : 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 Referenced by: §16.19 Permalink: http://dlmf.nist.gov/16.19.E6 Encodings: TeX, pMML, png See also: Annotations for §16.19 and Ch.16

where again $\vartheta=z\ifrac{\mathrm{d}}{\mathrm{d}z}$ . For conditions for (16.19.6) see Luke (1969a, Chapter 5). This reference and Mathai (1993, §§2.2 and 2.4) also supply additional identities.

16.18 Special Cases 16.20 Integrals and Series

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