Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (1.3)

\begin{equation*}
\gamma \left( s,u \right)=\frac{{{u}^{s}}}{s}M\left( s,s+1,-u \right).
\end{equation*} Proof:
\begin{align*}
& \gamma \left( s,u \right) \\
=&\int_{0}^{u}{{{x}^{s-1}}\exp (-x)dx} \\
=&\left[ \frac{{{x}^{s}}}{0!\left( s \right)}-\frac{{{x}^{s+1}}}{1!\left( s+1 \right)}+\frac{{{x}^{s+2}}}{2!\left( s+2 \right)}-\frac{{{x}^{s+3}}}{3!\left( s+3 \right)}+\cdots \right]{\Bigr|}_{0}^{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 \\
=&\frac{{{u}^{s}}}{s}\left[ 1+\frac{s}{\left( s+1 \right)}\frac{{{\left( -u \right)}^{1}}}{1!}+\frac{s\left( s+1 \right)}{\left( s+1 \right)\left( s+2 \right)}\frac{{{\left( -u \right)}^{2}}}{2!}+\frac{s\left( s+1 \right)\left( s+2 \right)}{\left( s+1 \right)\left( s+2 \right)\left( s+3 \right)}\frac{{{\left( -u \right)}^{3}}}{3!}+\cdots \right] \\
=&\frac{{{u}^{s}}}{s}M\left( s,s+1,-u \right).
\end{align*} $\square$

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