Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (4.12)

\begin{equation*}
{{Q}_{M}}\left(a,b\right)=\frac{\exp\left(\frac{-{{a}^{2}}}{2} \right)}{{{2}^{M}}\Gamma\left(M\right)}\sum\limits_{i=0}^{\infty}{{{\left( \frac{a}{2} \right)}^{2i}}}\frac{{{C}^{M+i}}{\exp\left({\tfrac{-C}{2}}\right)}h_{M-1+i}^{\tfrac{-C}{2}}-{{b}^{2\left(M+i \right)}}{\exp\left({\tfrac{-{{b}^{2}}}{2}}\right)}h_{M-1+i}^{\tfrac{-{{b}^{2}}}{2}}}{i!\prod\limits_{j=1}^{i}{\left( M-1+j \right)}}.
\end{equation*}Proof:
\begin{align*}
{{Q}_{M}}\left( a,b \right)&={{\int_{b}^{\infty }{x\left( \frac{x}{a} \right)}}^{M-1}}\exp \left( \frac{-\left( {{x}^{2}}+{{a}^{2}} \right)}{2} \right){{I}_{M-1}}\left( ax \right)dx \\
& =\frac{{{\left( \tfrac{1}{2} \right)}^{M-1}}\exp \left( \tfrac{-{{a}^{2}}}{2} \right)}{0!\Gamma \left( M \right)}\int_{b}^{\infty }{{{x}^{2M-1}}}\exp \left( \frac{-{{x}^{2}}}{2} \right)dx \\
& +\frac{{{\left( \tfrac{1}{2} \right)}^{M+1}}{{a}^{2}}\exp \left( \tfrac{-{{a}^{2}}}{2} \right)}{1!\Gamma \left( M+1 \right)}\int_{b}^{\infty }{{{x}^{2M+1}}}\exp \left( \frac{-{{x}^{2}}}{2} \right)dx \\
& +\frac{{{\left( \tfrac{1}{2} \right)}^{M+3}}{{a}^{4}}\exp \left( \tfrac{-{{a}^{2}}}{2} \right)}{2!\Gamma \left( M+2 \right)}\int_{b}^{\infty }{{{x}^{2M+3}}}\exp \left( \frac{-{{x}^{2}}}{2} \right)dx\\
&+\cdots \\
& =\frac{{{\left( \tfrac{1}{2} \right)}^{M-1}}\exp \left( \tfrac{-{{a}^{2}}}{2} \right)}{0!\Gamma \left( M \right)}\frac{1}{2}\left( {{u}^{M}}\exp \left( \frac{-u}{2} \right)h_{M-1}^{\tfrac{-u}{2}} \right){\Bigr|}_{{{b}^{2}}}^{\infty } \\
& +\frac{{{\left( \tfrac{1}{2} \right)}^{M+1}}{{a}^{2}}\exp \left( \tfrac{-{{a}^{2}}}{2} \right)}{1!\Gamma \left( M+1 \right)}\frac{1}{2}\left( {{u}^{M+1}}\exp \left( \frac{-u}{2} \right)h_{M}^{\tfrac{-u}{2}} \right){\Bigr|}_{{{b}^{2}}}^{\infty } \\
& +\frac{{{\left( \tfrac{1}{2} \right)}^{M+3}}{{a}^{4}}\exp \left( \tfrac{-{{a}^{2}}}{2} \right)}{2!\Gamma \left( M+2 \right)}\frac{1}{2}\left( {{u}^{M+2}}\exp \left( \frac{-u}{2} \right)h_{M+1}^{\tfrac{-u}{2}} \right){\Bigr|}_{{{b}^{2}}}^{\infty } \\
& +\cdots \\
& =\frac{\exp \left( \frac{-{{a}^{2}}}{2} \right)}{{{2}^{M}}\Gamma \left( M \right)}\sum\limits_{i=0}^{\infty }{{{\left( \frac{a}{2} \right)}^{2i}}}\frac{{{C}^{M+i}}{\exp\left({\tfrac{-C}{2}}\right)}h_{M-1+i}^{\tfrac{-C}{2}}-{{b}^{2\left( M+i\right)}}{\exp\left({\tfrac{-{{b}^{2}}}{2}}\right)}h_{M-1+i}^{\tfrac{-{{b}^{2}}}{2}}}{i!\prod\limits_{j=1}^{i}{\left( M-1+j \right)}},
\end{align*}
where $u=x^{2}$.
$\square$

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