# §16.19 Identities

 16.19.1 $\displaystyle{G^{m,n}_{p,q}}\left(\frac{1}{z};{a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}}\right)$ $\displaystyle={G^{n,m}_{q,p}}\left(z;{1-b_{1},\dots,1-b_{q}\atop 1-a_{1},\dots% ,1-a_{p}}\right),$ 16.19.2 $\displaystyle z^{\mu}{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}}\right)$ $\displaystyle={G^{m,n}_{p,q}}\left(z;{a_{1}+\mu,\dots,a_{p}+\mu\atop b_{1}+\mu% ,\dots,b_{q}+\mu}\right),$ 16.19.3 $\displaystyle{G^{m,n+1}_{p+1,q+1}}\left(z;{a_{0},\dots,a_{p}\atop b_{1},\dots,% b_{q},a_{0}}\right)$ $\displaystyle={G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}% }\right),$
 16.19.4 ${G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=\frac% {2^{p+1+b_{1}+\dots+b_{q}-m-n-a_{1}-\dots-a_{p}}}{\pi^{m+n-\frac{1}{2}(p+q)}}{% G^{2m,2n}_{2p,2q}}\left(2^{2p-2q}z^{2};{\frac{1}{2}a_{1},\frac{1}{2}a_{1}+% \frac{1}{2},\dots,\frac{1}{2}a_{p},\frac{1}{2}a_{p}+\frac{1}{2}\atop\frac{1}{2% }b_{1},\frac{1}{2}b_{1}+\frac{1}{2},\dots,\frac{1}{2}b_{q},\frac{1}{2}b_{q}+% \frac{1}{2}}\right),$
 16.19.5 $\vartheta{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}% \right)={G^{m,n}_{p,q}}\left(z;{a_{1}-1,a_{2},\dots,a_{p}\atop b_{1},\dots,b_{% q}}\right)+(a_{1}-1){G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots% ,b_{q}}\right),$
 16.19.6 $\int_{0}^{1}t^{-a_{0}}(1-t)^{a_{0}-b_{q+1}-1}{G^{m,n}_{p,q}}\left(zt;{a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}}\right)\mathrm{d}t=\Gamma\left(a_{0}-b_{q+1% }\right){G^{m,n+1}_{p+1,q+1}}\left(z;{a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q+% 1}}\right),$

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.