\begin{definition}[Definition:Gamma Function/Partial]
Let $m \in \Z_{\ge 0}$.
The '''partial gamma function at $m$''' is defined as:
:$\ds \map {\Gamma_m} z := \frac {m^z m!} {z \paren {z + 1} \paren {z + 2} \cdots \paren {z + m} }$
which is valid except for $z \in \set {0, -1, -2, \ldots, -m}$.
\end{definition}