Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (3.13)

\begin{equation*}
\left( s+1 \right)h_{s}^{c}=1-ch_{s+1}^{c}.
\end{equation*}Proof:
\begin{align*}
& h_{s}^{c}=\frac{1}{\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 \\
& \left( s+1 \right)h_{s}^{c}=1-c\left[ \frac{1}{\left( s+2 \right)}+\frac{{{\left( -c \right)}^{1}}}{\left( s+2 \right)\left( s+3 \right)}+\frac{{{\left( -c \right)}^{2}}}{\left( s+2 \right)\left( s+3 \right)\left( s+4 \right)}+\cdots \right] \\
& \left( s+1 \right)h_{s}^{c}=1-ch_{s+1}^{c}.
\end{align*}
$\square$

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