Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (2.12)

$$\begin{equation*} \Gamma \left( -r+1 \right)=-r\Gamma \left( -r \right) \end{equation*}$$ Proof:
$$\begin{align*} \Gamma \left( -r+1 \right) &={{c}^{-r+1}}\exp (-c)h_{-r}^{-c}{{|}_{c\to \infty }} \\ & ={{c}^{-r+1}}\exp (-c)\left\{ -{{c}^{-1}}-r{{c}^{r-1}}\exp (c)\Gamma \left( -r \right) \right\}{{|}_{c\to \infty }} \\ & =-{{c}^{-r}}\exp (-c)-r\Gamma \left( -r \right){{|}_{c\to \infty }} \\   & =-r\Gamma \left( -r \right) \\ \end{align*}$$
$\square$

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