Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (4.9)

$$\begin{equation*} B\left(\alpha,\beta\right)=\frac{{\exp(-C)}h_{\alpha-1}^{-C}h_{\beta -1}^{-C}}{h_{\alpha +\beta -1}^{-C}}. \end{equation*}$$ Proof:
$$\begin{align*} B\left( \alpha ,\beta \right) &=\frac{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}{\Gamma \left( \alpha +\beta \right)} \\ & =\frac{{{C}^{\alpha }}\exp \left( -C \right)h_{\alpha -1}^{-C}{{C}^{\beta }}\exp \left( -C \right)h_{\beta -1}^{-C}}{{{C}^{\alpha +\beta }}\exp \left( -C \right)h_{\alpha +\beta -1}^{-C}} \\ & =\frac{\exp (-C)h_{\alpha -1}^{-C}h_{\beta -1}^{-C}}{h_{\alpha +\beta -1}^{-C}}. \end{align*}$$
$\square$

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