Indefinite Integration of the Gamma Integral
and Related Statistical Applications
Definition (3.1)
If $s\notin {{\mathbb{Z}}^{-}}$, \begin{equation*} 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. \end{equation*}If $s=-1$, \begin{equation*} h_{s}^{c}=\exp (-c)\left( \log \left| c \right|+h_{0}^{-c}+h_{1}^{-c}+h_{2}^{-c}+h_{3}^{-c}+\cdots \right). \end{equation*}
If $s\in {{\mathbb{Z}}^{-}}$ & $s\ne -1$, \begin{equation*} 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)}+\cdots +\frac{{{\left( -c \right)}^{-s-1}}}{\prod\limits_{j=0}^{-s-2}{\left( s+1+j \right)}}h_{-1}^{c}. \end{equation*}
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