\[\left({\textstyle\ifrac{\prod\limits_{k=1}^{q+1}\Gamma\left(a_{k}\right)}{%
\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}}\right){{}_{q+1}F_{q}}\left({a%
_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=\sum_{j=1}^{q+1}\left(z_{0%
}-z\right)^{-a_{j}}\sum_{n=0}^{\infty}\frac{\Gamma\left(a_{j}+n\right)}{n!}\*%
\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}k=1\\
k\neq j\end{subarray}}^{q+1}\Gamma\left(a_{k}-a_{j}-n\right)}{\prod\limits_{k=%
1}^{q}\Gamma\left(b_{k}-a_{j}-n\right)}}\right)\*{{}_{q+1}F_{q}}\left({a_{1}-a%
_{j}-n,\dots,a_{q+1}-a_{j}-n\atop b_{1}-a_{j}-n,\dots,b_{q}-a_{j}-n};z_{0}%
\right)\left(z-z_{0}\right)^{-n}.\]