\[\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}U\left(a,b,z\right)%
\right)=(-1)^{n}{\left(a-b+1\right)_{n}}z^{b-n-1}U\left(a,b-n,z\right),\]