Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (4.6)

\begin{align*}
{{k}_{2}}>&{{k}_{1}}+\left( \lambda -1 \right){\exp(-C\lambda )}\left[ 1+\left( \lambda -1 \right)\left( C-1 \right)h_{1}^{-C\left( \lambda -1 \right)} \right] \notag \\
& +\sum\limits_{i=0}^{\infty }{{{\left[ C\left( \lambda -1 \right) \right]}^{i+1}}{\exp\left(-C\left( \lambda -1 \right)\right)}h_{t+1+i}^{C}h_{i}^{-C\left( \lambda -1 \right)}}.
\end{align*}Proof:
According to (4.5),
\begin{equation*}
{{k}_{2}}={{E}_{\left( 1 \right)}}\left( x;\lambda C,t \right)=\log \left| \frac{\lambda C}{x} \right|+\sum\limits_{s=0}^{t}{\left( h_{s}^{\lambda C}-h_{s}^{x} \right)}.
\end{equation*}
Given the multiplication formula, we know
\begin{align*}
h_{s}^{\lambda C}&=h_{s}^{C\left( \lambda -1 \right)+C} \\
& =h_{s}^{C}+C\left( \lambda -1 \right)\frac{dh_{s}^{C}}{dC}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\frac{{{d}^{2}}h_{s}^{C}}{d{{C}^{2}}}+\cdots.
\end{align*}
Bringing $h_{s}^{\lambda C}$ back to ${{E}_{\left( 1 \right)}}\left( x;\lambda C,t \right)$, we derive
\begin{align*}
& {{E}_{\left( 1 \right)}}\left( x;\lambda C,t \right)=\log \left| \frac{\lambda C}{x} \right|+\sum\limits_{s=0}^{t}{\left\{ \left[ h_{s}^{C}+C\left( \lambda -1 \right)\frac{dh_{s}^{C}}{dC}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\frac{{{d}^{2}}h_{s}^{C}}{d{{C}^{2}}}+\cdots \right]-h_{s}^{x} \right\}} \\
& =\log \lambda +\log C-\log x+\sum\limits_{s=0}^{t}{\left[ h_{s}^{C}-h_{s}^{x} \right]}+\sum\limits_{s=0}^{t}{\left[ C\left( \lambda -1 \right)\frac{dh_{s}^{C}}{dC}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\frac{{{d}^{2}}h_{s}^{C}}{d{{C}^{2}}}+\cdots \right]}.
\end{align*}
We need to work out the last term.
\begin{align*}
& \sum\limits_{s=0}^{t}{\left[ C\left( \lambda -1 \right)\frac{dh_{s}^{C}}{dC}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\frac{{{d}^{2}}h_{s}^{C}}{d{{C}^{2}}}+\cdots \right]} \\
& =\left[ C\left( \lambda -1 \right) \right]\sum\limits_{s=0}^{t}{\frac{dh_{s}^{C}}{dC}}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\sum\limits_{s=0}^{t}{\frac{{{d}^{2}}h_{s}^{C}}{d{{C}^{2}}}}+\cdots \\
& =\left[ C\left( \lambda -1 \right) \right]\left[ \left( h_{1}^{C}-h_{0}^{C} \right)+\left( h_{2}^{C}-h_{1}^{C} \right)+\cdots +\left( h_{t+1}^{C}-h_{t}^{C} \right) \right] \\
& +\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\left[ \left( h_{2}^{C}-2h_{1}^{C}+h_{0}^{C} \right)+\left( h_{3}^{C}-2h_{2}^{C}+h_{1}^{C} \right)+\cdots +\left( h_{t+2}^{C}-2h_{t+1}^{C}+h_{t}^{C} \right) \right] \\
& +\cdots \\
& =\left[ C\left( \lambda -1 \right) \right]\left[ h_{t+1}^{C}-h_{0}^{C} \right]+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\left[ h_{t+2}^{C}-h_{t+1}^{C}-h_{1}^{C}+h_{0}^{C} \right]+\cdots \\
& =\left\{ -\left[ C\left( \lambda -1 \right) \right]h_{0}^{C}-\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\frac{dh_{0}^{C}}{dC}-\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{3}}}{3!}\frac{{{d}^{2}}h_{0}^{C}}{d{{C}^{2}}}-\cdots \right\} \\
& +\left\{ \left[ C\left( \lambda -1 \right) \right]h_{t+1}^{C}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\frac{dh_{t+1}^{C}}{dC}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{3}}}{3!}\frac{{{d}^{2}}h_{t+1}^{C}}{d{{C}^{2}}}+\cdots \right\}.
\end{align*}
We have already known
\begin{align*}
h_{0}^{C}=&-{{C}^{-1}}\exp \left( -C \right)+{{C}^{-1}} \\
\frac{dh_{0}^{C}}{dC}=&{{C}^{-2}}\exp \left( -C \right)+{{C}^{-1}}\exp \left( -C \right)-{{C}^{-2}} \\
\frac{{{d}^{2}}h_{0}^{C}}{d{{C}^{2}}}=&-2{{C}^{-3}}\exp \left( -C \right)-2{{C}^{-2}}\exp \left( -C \right)-{{C}^{-1}}\exp \left( -C \right)+2{{C}^{-3}} \\
& \vdots
\end{align*}
and hence, we can work out the last two terms, respectively.
\begin{align*}
& -\left[ C\left( \lambda -1 \right) \right]h_{0}^{C}-\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\frac{dh_{0}^{C}}{dC}-\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{3}}}{3!}\frac{{{d}^{2}}h_{0}^{C}}{d{{C}^{2}}}-\cdots \\
& =\left( \lambda -1 \right)\left[ \exp \left( -C \right)-1 \right] \\
& +{{\left( \lambda -1 \right)}^{2}}\left[ \frac{-\exp \left( -C \right)}{2!}+\frac{-C\exp \left( -C \right)}{2!}+\frac{1}{2!} \right] \\
& +{{\left( \lambda -1 \right)}^{3}}\left[ \frac{2\exp \left( -C \right)}{3!}+\frac{2C\exp \left( -C \right)}{3!}+\frac{{{C}^{2}}\exp \left( -C \right)}{3!}-\frac{2!}{3!} \right] \\
& +\cdots \\
& =-\log \lambda +\frac{\exp \left( -C \right)}{0!}\log \lambda +\frac{C\exp \left( -C \right)}{1!}\left[ \log \lambda -\frac{{{\left( k-1 \right)}^{1}}}{1} \right] \\
& +\frac{{{C}^{2}}\exp \left( -C \right)}{2!}\left[ \log \lambda -\frac{{{\left( k-1 \right)}^{1}}}{1}+\frac{{{\left( k-1 \right)}^{2}}}{2} \right]+\cdots \\
& =-\log \lambda +\frac{\exp \left( -C \right)}{0!}\log \lambda +\frac{C\exp \left( -C \right)}{1!}\left[ \log \lambda +\frac{{{\left( 1-k \right)}^{1}}}{1} \right] \\
& +\frac{{{C}^{2}}\exp \left( -C \right)}{2!}\left[ \log \lambda +\frac{{{\left( 1-k \right)}^{1}}}{1}+\frac{{{\left( 1-k \right)}^{2}}}{2} \right]+\cdots,
\end{align*}
\begin{align*}
& \left[ C\left( \lambda -1 \right) \right]h_{t+1}^{C}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\frac{dh_{t+1}^{C}}{dC}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{3}}}{3!}\frac{{{d}^{2}}h_{t+1}^{C}}{d{{C}^{2}}}+\cdots \\
& =\left[ C\left( \lambda -1 \right) \right]h_{t+1}^{C}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{2!}\left( h_{t+2}^{C}-h_{t+1}^{C} \right) \\
& +\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{3}}}{3!}\left( h_{t+3}^{C}-2h_{t+2}^{C}+h_{t+1}^{C} \right)+\cdots \\
& =h_{t+1}^{C}\left[ \frac{{{\left[ C\left( \lambda -1 \right) \right]}^{1}}}{0!\cdot 1}-\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{1!\cdot 2}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{3}}}{2!\cdot 3}-\cdots \right] \\
& +h_{t+2}^{C}\left[ \frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{0!\cdot 2}-\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{3}}}{1!\cdot 3}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{4}}}{2!\cdot 4}-\cdots \right] \\
& +h_{t+3}^{C}\left[ \frac{{{\left[ C\left( \lambda -1 \right) \right]}^{3}}}{0!\cdot 3}-\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{4}}}{1!\cdot 4}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{5}}}{2!\cdot 5}-\cdots \right] \\
& +\cdots \\
& =h_{t+1}^{C}\int_{0}^{C\left( \lambda -1 \right)}{{\exp(-u)}du}+h_{t+2}^{C}\int_{0}^{C\left( \lambda -1 \right)}{{{u}^{1}}{\exp(-u)}du}+h_{t+3}^{C}\int_{0}^{C\left( \lambda -1 \right)}{{{u}^{2}}{\exp(-u)}du}+\cdots \\
& =h_{t+1}^{C}{{u}^{1}}\exp \left( -u \right)h_{0}^{-u}+h_{t+2}^{C}{{u}^{2}}\exp \left( -u \right)h_{1}^{-u}+h_{t+3}^{C}{{u}^{3}}\exp \left( -u \right)h_{2}^{-u}+\cdots{{|}_{u=C\left( \lambda -1 \right)}}.
\end{align*}
For the first term, we can further specify its lower limit, since
\begin{align*}
\log \lambda >&\frac{{{\left( \lambda -1 \right)}^{1}}}{1}-\frac{{{\left( \lambda -1 \right)}^{2}}}{2} \\
-\log \lambda +\frac{{{\left( \lambda -1 \right)}^{1}}}{1}>&\frac{{{\left( \lambda -1 \right)}^{2}}}{2}-\frac{{{\left( \lambda -1 \right)}^{3}}}{3} \\
& \vdots
\end{align*}
and therefore,
\begin{align*}
& -\log \lambda +\frac{\exp \left( -C \right)}{0!}\log \lambda +\frac{C\exp \left( -C \right)}{1!}\left[ \log \lambda +\frac{{{\left( 1-\lambda \right)}^{1}}}{1} \right] \\
& +\frac{{{C}^{2}}\exp \left( -C \right)}{2!}\left[ \log \lambda +\frac{{{\left( 1-\lambda \right)}^{1}}}{1}+\frac{{{\left( 1-\lambda \right)}^{2}}}{2} \right]+\cdots \\
& >-\log \lambda +\frac{{{\left( -C \right)}^{0}}\exp \left( -C \right)}{0!}\left[ \frac{{{\left( \lambda -1 \right)}^{1}}}{1}-\frac{{{\left( \lambda -1 \right)}^{2}}}{2} \right] \\
& +\frac{{{\left( -C \right)}^{1}}\exp \left( -C \right)}{1!}\left[ \frac{{{\left( \lambda -1 \right)}^{2}}}{2}-\frac{{{\left( \lambda -1 \right)}^{3}}}{3} \right] \\
& +\frac{{{\left( -C \right)}^{2}}\exp \left( -C \right)}{2!}\left[ \frac{{{\left( \lambda -1 \right)}^{3}}}{3}-\frac{{{\left( \lambda -1 \right)}^{4}}}{4} \right] \\
& +\cdots \\
& =-\log \lambda +\frac{\exp \left( -C \right)}{C}\left\{ \frac{{{\left[ C\left( \lambda -1 \right) \right]}^{1}}}{0!\cdot 1}-\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{1!\cdot 2}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{3}}}{2!\cdot 3}-\cdots \right\} \\
& -\frac{\exp \left( -C \right)}{{{C}^{2}}}\left\{ \frac{{{\left[ C\left( \lambda -1 \right) \right]}^{1}}}{0!\cdot 2}-\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{2}}}{1!\cdot 3}+\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{3}}}{2!\cdot 4}-\cdots \right\} \\
& =-\log \lambda +{{C}^{-1}}\exp \left( -C \right)\int_{0}^{C\left( \lambda -1 \right)}{\exp \left( -u \right)du}-{{C}^{-2}}\exp \left( -C \right)\int_{0}^{C\left( \lambda -1 \right)}{u\exp \left( -u \right)du} \\
& =-\log \lambda +{{C}^{-1}}\exp \left( -C-u \right){{u}^{1}}h_{0}^{-u}-{{C}^{-2}}\exp \left( -C-u \right){{u}^{2}}h_{1}^{-u}{{|}_{u=C\left( \lambda -1 \right)}}.
\end{align*}
We can bring these two terms back to ${{k}_{2}}$
\begin{align}
{{k}_{2}}&={{E}_{\left( 1 \right)}}\left( x;\lambda C,t \right)-{{E}_{\left( 1 \right)}}\left( x \right) \notag \\
& =\log \lambda +\log C-\log x+\sum\limits_{s=0}^{t}{\left( h_{s}^{C}-h_{s}^{x} \right)}+\sum\limits_{s=0}^{t}{\left( \sum\limits_{i=1}^{\infty }{\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{i}}}{i!}\frac{{{d}^{\left( i \right)}}h_{s}^{C}}{d{{C}^{i}}}} \right)}-{{E}_{\left( 1 \right)}}\left( x \right) \notag \\
& ={{k}_{1}}+\log \lambda -\sum\limits_{i=1}^{\infty }{\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{i}}}{i!}}\frac{{{d}^{\left( i-1 \right)}}h_{0}^{C}}{d{{C}^{i-1}}}+\sum\limits_{i=1}^{\infty }{\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{i}}}{i!}}\frac{{{d}^{\left( i-1 \right)}}h_{t+1}^{C}}{d{{C}^{i-1}}} \notag \\
& ={{k}_{1}}+\sum\limits_{i=0}^{\infty }{\frac{{{C}^{i}}{\exp(-C)}}{i!}\left( \log \lambda +\sum\limits_{j=1}^{i}{\frac{{{\left( 1-\lambda \right)}^{j}}}{j}} \right)}+\sum\limits_{i=1}^{\infty }{\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{i}}}{i!}}\frac{{{d}^{\left( i-1 \right)}}h_{t+1}^{C}}{d{{C}^{i-1}}} \notag \\
& >{{k}_{1}}+\sum\limits_{i=0}^{\infty }{\left( \frac{{{\left( -C \right)}^{i}}{\exp(-C)}}{i!} \right)}\left( \frac{{{\left( \lambda -1 \right)}^{i+1}}}{\left( i+1 \right)}-\frac{{{\left( \lambda -1 \right)}^{i+2}}}{\left( i+2 \right)} \right)+\sum\limits_{i=1}^{\infty }{\frac{{{\left[ C\left( \lambda -1 \right) \right]}^{i}}}{i!}}\frac{{{d}^{\left( i-1 \right)}}h_{t+1}^{C}}{d{{C}^{i-1}}} \notag \\
&={{k}_{1}}+{{C}^{-1}}u\exp (-C-u)\left( h_{0}^{-u}-{{C}^{-1}}uh_{1}^{-u} \right)+\sum\limits_{i=0}^{\infty }{{{u}^{i+1}}{\exp(-u)}h_{i}^{-u}h_{t+1+i}^{C}}{{\Bigr|}_{u=C\left( \lambda -1 \right)}}, \notag
\end{align}
where all the three terms in the RHS equation are positive.

$\square$

Welcome all math lovers!