§16.19 Identities

Notes:: All of the results in this section are straightforward consequences of the definition (16.17.1).
Keywords:: Meijer $G$ -function
Permalink:: http://dlmf.nist.gov/16.19

16.19.1 $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(\frac{1}{z};{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\mathop{{G^{{n,m}}_{{q,p}}}\/}\nolimits\!\left(z;{1-b_{1},\dots,1-b_{q}\atop 1-a_{1},\dots,1-a_{p}}\right),$

Symbols:: $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)$ : 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

16.19.2 $z^{\mu}\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1}+\mu,\dots,a_{p}+\mu\atop b_{1}+\mu,\dots,b_{q}+\mu}\right),$

Symbols:: $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)$ : 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

16.19.3 $\mathop{{G^{{m,n+1}}_{{p+1,q+1}}}\/}\nolimits\!\left(z;{a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q},a_{0}}\right)=\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right),$

Symbols:: $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)$ : 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

16.19.4 $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\frac{2^{{p+1+b_{1}+\dots+b_{q}-m-n-a_{1}-\dots-a_{p}}}}{\pi^{{m+n-\frac{1}{2}(p+q)}}}\mathop{{G^{{2m,2n}}_{{2p,2q}}}\/}\nolimits\!\left(2^{{2p-2q}}z^{2};{\frac{1}{2}a_{1},\frac{1}{2}a_{1}+\frac{1}{2},\dots,\frac{1}{2}a_{p},\frac{1}{2}a_{p}+\frac{1}{2}\atop\frac{1}{2}b_{1},\frac{1}{2}b_{1}+\frac{1}{2},\dots,\frac{1}{2}b_{q},\frac{1}{2}b_{q}+\frac{1}{2}}\right),$

Symbols:: $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)$ : 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.E4
Encodings:: TeX, pMML, png

16.19.5 $\vartheta\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1}-1,a_{2},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)+(a_{1}-1)\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right),$

Symbols:: $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)$ : 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

16.19.6 $\int _{0}^{1}t^{{-a_{0}}}(1-t)^{{a_{0}-b_{{q+1}}-1}}\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(zt;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)dt=\mathop{\Gamma\/}\nolimits\!\left(a_{0}-b_{{q+1}}\right)\mathop{{G^{{m,n+1}}_{{p+1,q+1}}}\/}\nolimits\!\left(z;{a_{0},\dots,a_{p}\atop b_{1},\dots,b_{{q+1}}}\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)$ : Meijer $G$ -function, $dx$ : 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

where again $\vartheta=z\ifrac{d}{dz}$ . 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.