Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (3.9)

\begin{equation*}
\frac{{{\partial }^{n}}\left( {\exp(c)}h_{s}^{c} \right)}{\partial {{c}^{n}}}={\exp(c)}h_{s+n}^{c}.
\end{equation*}Proof:
For the first-order partial derivative of ${\exp(c)}h_{s}^{c}$,
\begin{align*}
\frac{\partial \left( \exp (c)h_{s}^{c} \right)}{\partial c}&=\exp (c)\frac{\partial \left( h_{s}^{c} \right)}{\partial c}+\exp (c)h_{s}^{c} \\
& =\exp (c)\left( h_{s+1}^{c}-h_{s}^{c} \right)+\exp (c)h_{s}^{c} \\
& =\exp (c)h_{s+1}^{c}.
\end{align*}
Repeating the first-order partial derivative $n$ times, we can derive
\begin{equation*}
\frac{{{\partial }^{n}}\left( {\exp(c)}h_{s}^{c} \right)}{\partial {{c}^{n}}}={\exp(c)}h_{s+n}^{c}.
\end{equation*}
$\square$

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