Indefinite Integration of the Gamma Integral
and Related Statistical Applications
Definition of Equation (1.5)
\begin{equation*}h_{s}^{c}\equiv\frac{1-M\left( 1,s+1,-c \right)}{c}.
\end{equation*}
Proof:
\begin{align*}
h_{s}^{c}\equiv &\frac{1-M\left( 1,s+1,-c \right)}{c} \\
=&\frac{1}{c}\left\{ 1-1-\frac{1}{s+1}\cdot \left( -c \right)-\frac{1\cdot 2}{\left( s+1 \right)\left( s+2 \right)}\cdot \frac{{{\left( -c \right)}^{2}}}{2!}-\frac{1\cdot 2\cdot 3}{\left( s+1 \right)\left( s+2 \right)\left( s+3 \right)}\cdot \frac{{{\left( -c \right)}^{3}}}{3!} \right\} \\
=&\frac{{{\left( -c \right)}^{0}}}{s+1}+\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)}+\cdotsÂ
\end{align*}$\square$
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