\[{{}_{p+1}F_{p}}\left({a_{1}+r,\dots,a_{k-1}+r,a_{k},\dots,a_{p+1}\atop b_{1}+r%
,\dots,b_{k}+r,b_{k+1},\dots,b_{p}};z\right)=\sum_{n=0}^{m-1}\frac{{\left(a_{1%
}+r\right)_{n}}\cdots{\left(a_{k-1}+r\right)_{n}}{\left(a_{k}\right)_{n}}%
\cdots{\left(a_{p+1}\right)_{n}}}{{\left(b_{1}+r\right)_{n}}\cdots{\left(b_{k}%
+r\right)_{n}}{\left(b_{k+1}\right)_{n}}\cdots{\left(b_{p}\right)_{n}}}\frac{z%
^{n}}{n!}+O\left(\frac{1}{r^{m}}\right),\]