Definition:Gamma Function/Partial

Definition

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 $z \in \set {0, -1, -2, \ldots, -m}$.

The term partial gamma function was coined by $\mathsf{Pr} \infty \mathsf{fWiki}$ 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.