Expressing Kummer U function as Meijer G

Question

I want to express $U(n,m+n+1,z)$, where $n \in \mathbb{N}$, $m \in \mathbb{N}^+$, $n,m < \infty$ and $z > 0$ in term of Meijer's G-function. Wolfram provides the following equation $$ U(a,b,z) = \frac{1}{\Gamma(a)\Gamma(a-b+1)} G^{2,1}_{1,2}\left(\left. \begin{matrix} 1-a\\0,1-b \end{matrix}\right| z \right), $$ You can check it through: http://functions.wolfram.com/07.33.26.0004.01. However when I used this equation for my need, it produces $$ U(n,n+m+1,z)= \frac{1}{\Gamma(n)\Gamma(-m)} G^{2,1}_{1,2}\left(\left. \begin{matrix} 1-n\\0,-n-m \end{matrix}\right| z \right), $$ which is not valid (the Gamma and Meijer's G terms) according to what I've known.

Answer 1

The last formula is valid as the limit for $m$ approaching an integral value. For your values of the parameters, the function becomes a finite sum: $$U(n, n + m + 1, z) = \frac {m! \hspace {1px} z^{-n}} {(n - 1)!} \sum_{k = 0}^m \frac {(n + k - 1)!} {(m - k)!} \frac {z^{-k}} {k!}.$$ There are some contrived ways to write a power function as a G-function; they can probably be extended to sums of powers.