Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (3.6)

$$\begin{equation*} h_{s}^{p{{q}^{-1}}}=h_{s}^{p}+p\left( 1-{{q}^{-1}} \right)\sum\limits_{i=0}^{\infty }{\frac{{{\left( -p{q}^{-1} \right)}^{i}}}{\prod\limits_{j=0}^{i}{\left( s+1+i \right)}}{{h}_{s+1+i}^{p}}}. \end{equation*}$$ Proof:
Replacing $q$ with $q^{-1}$ in the multiplication formula (3.5), we can derive the division formula.

$\square$

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