Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (3.3)

\begin{equation*}
h_{s}^{p+q}=\sum\limits_{i=0}^{\infty }{\frac{{{q}^{i}}}{i!}\frac{{{\partial }^{\left( i \right)}}h_{s}^{p}}{\partial {{p}^{i}}}}.
\end{equation*} Proof:
\begin{align*}
h_{s}^{p+q}&=\frac{1}{\left( s+1 \right)}-\frac{\left( p+q \right)}{\left( s+1 \right)\left( s+2 \right)}+\frac{{{\left( p+q \right)}^{2}}}{\left( s+1 \right)\left( s+2 \right)\left( s+3 \right)}+\cdots \\
& =h_{s}^{p}+\frac{-q}{\left( s+1 \right)\left( s+2 \right)}+\frac{2pq+{{q}^{2}}}{\left( s+1 \right)\left( s+2 \right)\left( s+3 \right)}+\frac{-3{{p}^{2}}q-3p{{q}^{2}}-{{q}^{3}}}{\left( s+1 \right)\left( s+2 \right)\left( s+3 \right)\left( s+3 \right)} \\
& =h_{s}^{p}+\frac{{{q}^{1}}}{1!}\left[ \frac{-1}{\left( s+1 \right)\left( s+2 \right)}+\frac{2p}{\left( s+1 \right)\left( s+2 \right)\left( s+3 \right)}+\cdots \right] \\
& +\frac{{{q}^{2}}}{2!}\left[ \frac{2!}{\left( s+1 \right)\left( s+2 \right)\left( s+3 \right)}+\frac{-6p}{\left( s+1 \right)\left( s+2 \right)\left( s+3 \right)\left( s+4 \right)}+\cdots \right] \\
& =\frac{{{q}^{0}}}{0!}\frac{{{\partial }^{\left( 0 \right)}}h_{s}^{p}}{\partial {{p}^{0}}}+\frac{{{q}^{1}}}{1!} \frac{{{\partial }^{\left( 1 \right)}}h_{s}^{p}}{\partial {{p}^{1}}}+\frac{{{q}^{2}}}{2!}\frac{{{\partial }^{\left( 2 \right)}}h_{s}^{p}}{\partial {{p}^{2}}}+\cdots \\
& =\sum\limits_{i=0}^{\infty }{\frac{{{q}^{i}}}{i!}\frac{{{\partial }^{\left( i \right)}}h_{s}^{p}}{\partial {{p}^{i}}}}.
\end{align*}
$\square$

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