\[U\left(a,n+1,z\right)=\frac{(-1)^{n+1}}{n!\Gamma\left(a-n\right)}\sum_{k=0}^{%
\infty}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+%
\psi\left(a+k\right)-\psi\left(1+k\right)-\psi\left(n+k+1\right)\right)+\frac{%
1}{\Gamma\left(a\right)}\sum_{k=1}^{n}\frac{(k-1)!{\left(1-a+k\right)_{n-k}}}{%
(n-k)!}z^{-k}.\]