Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (3.5)

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

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