\[\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}%
\left(a,b;-n;z\right)=\frac{{\left(a\right)_{n+1}}{\left(b\right)_{n+1}}}{(n+1%
)!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z\right),\]