Definition:Gamma Function/Partial

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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

Linguistic Note

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.