Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (2.2)

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

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