\[{{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}%
{k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}%
{{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},\]