Definition:Gamma Function/Partial

Definition
Let $m \in \Z_{\ge 0}$.

The partial Gamma function at $m$ is defined as:
 * $\displaystyle \Gamma_m \left({z}\right) := \frac {m^z m!} {z \left({z + 1}\right) \left({z + 2}\right) \cdots \left({z + m}\right)}$

which is valid except for $z \in \left\{{0, -1, -2, \ldots, -m}\right\}$.

Also see

 * Definition:Gamma Function

Linguistic Note
The term partial Gamma function was coined by as a convenient term to identify this concept.

It is not to be confused with the incomplete Gamma function, which is a completely different thing.