Indefinite Integration of the Gamma Integral
and Related Statistical Applications
Identities of Equation (2.1)
\begin{align*}
h_{s}^{c}&=\frac{{{\left( -c \right)}^{0}}}{\left( s+1 \right)}+\frac{{{\left( -c \right)}^{1}}}{\left( s+1 \right)\left( s+2
\right)}+\frac{{{\left( -c \right)}^{2}}}{\left( s+1 \right)\left( s+2 \right)\left( s+3 \right)}+\frac{{{\left( -c \right)}^{3}}}{\left( s+1 \right)\left( s+2 \right)\left( s+3 \right)\left( s+4 \right)}+\cdots \\
&=\frac{1}{s+1}\left( 1-\frac{c}{s+2}\left( 1-\frac{c}{s+3}\left( 1-\frac{c}{s+4}\left( \cdots \right) \right) \right) \right) \\
&=\frac{{{\left( -c \right)}^{0}}}{0!}\left( \frac{C_{0}^{0}}{s+1} \right)+\frac{{{\left( -c \right)}^{1}}}{1!}\left(
\frac{C_{0}^{1}}{s+1}-\frac{C_{1}^{1}}{s+2} \right)+\frac{{{\left( -c \right)}^{2}}}{2!}\left(
\frac{C_{0}^{2}}{s+1}-\frac{C_{1}^{2}}{s+2}+\frac{C_{2}^{2}}{s+3} \right)+\cdots \\
&=\exp \left( -c \right)\left\{ \frac{{{c}^{0}}}{0!\left( s+1 \right)}+\frac{{{c}^{1}}}{1!\left( s+2
\right)}+\frac{{{c}^{2}}}{2!\left( s+3 \right)}+\frac{{{c}^{3}}}{3!\left( s+4 \right)}+\cdots \right\} \\
&=\frac{1}{s+1}-\frac{c}{s+1}h_{s+1}^{c}
\end{align*}
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