# §16.19 Identities

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

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.