Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (1.2)

$$\begin{equation*} g(s,-1,u)=\frac{{{u}^{s}}}{0!\left( s \right)}-\frac{{{u}^{s+1}}}{1!\left( s+1 \right)}+\frac{{{u}^{s+2}}}{2!\left( s+2 \right)}-\frac{{{u}^{s+3}}}{3!\left( s+3 \right)}+\cdots. \end{equation*}$$ Proof:
$$\begin{align*} g(s,-1,u) =&\int{{{u}^{s-1}}\exp \left( -u \right)du} \\ =&\int{\left( \frac{{{u}^{s-1}}}{0!}-\frac{{{u}^{s}}}{1!}+\frac{{{u}^{s+1}}}{2!}-\frac{{{u}^{s+2}}}{3!}+\cdots \right)}du \\ =&\frac{{{u}^{s}}}{0!\left( s \right)}-\frac{{{u}^{s+1}}}{1!\left( s+1 \right)}+\frac{{{u}^{s+2}}}{2!\left( s+2 \right)}-\frac{{{u}^{s+3}}}{3!\left( s+3 \right)}+\cdots. \end{align*}$$ $\square$

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