Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (4.4)

\begin{equation*}
{{E}_{\left( 1 \right)}}\left( x;C,t \right)-{{E}_{\left( 1 \right)}}\left( x;C \right)=\underset{n\to \infty }{\mathop{\lim }}\,\,\,\sum\limits_{s=t+1}^{n}{\left( h_{s}^{x}-h_{s}^{C} \right)}.
\end{equation*}Proof:
\begin{align*}
& {{E}_{\left( 1 \right)}}\left( x;C,t \right)-{{E}_{\left( 1 \right)}}\left( x;C \right) \\
& =\left[ \left( \log \left| -C \right|+\,\sum\limits_{s=0}^{t}{h_{s}^{C}} \right)-\left( \log \left| -x \right|+\,\sum\limits_{s=0}^{t}{h_{s}^{x}} \right) \right] \\
& -\left[ \left( \log \left| -C \right|+\underset{n\to \infty }{\mathop{\lim }}\,\,\sum\limits_{s=0}^{n}{h_{s}^{C}} \right)-\left( \log \left| -x \right|+\underset{n\to \infty }{\mathop{\lim }}\,\,\sum\limits_{s=0}^{n}{h_{s}^{x}} \right) \right] \\
& =\,\sum\limits_{s=0}^{t}{h_{s}^{C}}-\underset{n\to \infty }{\mathop{\lim }}\,\,\sum\limits_{s=0}^{n}{h_{s}^{C}}-\sum\limits_{s=0}^{t}{h_{s}^{x}}+\underset{n\to \infty }{\mathop{\lim }}\,\,\sum\limits_{s=0}^{n}{h_{s}^{x}} \\
& =\underset{n\to \infty }{\mathop{\lim }}\,\,\,\sum\limits_{s=t+1}^{n}{\left( h_{s}^{x}-h_{s}^{C} \right)}.
\end{align*}
$\square$

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