Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (3.7)

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

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