Let $Y$ be a beta-distributed random variable, such that $Y \sim \mathrm{Beta}(N,LN-N)$. Is there any closed-form expression for the inverse moments of $Y$, i.e. \begin{equation*} E\{Y^{-\alpha}\},\quad \alpha \in \mathbb{N}? \end{equation*} In particular, I'm interested in $E\{Y^{-3}\}$ and $E\{Y^{-4}\}$.
Thanks!
Let $Y\sim \text{Beta}(p,q)$. Then $$ Y=_{d}\frac{Z}{Z+W}\tag{1} $$ for some $Z\sim \text{Gamma}(p)$ and $W\sim \text{Gamma}(q)$ where $Z$ and W are independent ($p,q>0$). Further $Y$ is independent of the denominator $T=Z+W\sim \text{Gamma} (p+q)$. Here $=_{d}$ means equal in distribution and $Z\sim \text{Gamma}(p)$ means that $Z$ has density $$ f_{Z}(z)=\frac{1}{\Gamma(p)}z^{p-1}e^{-z}\quad (z>0). $$ In particular $$ EZ^d=\Gamma(p+d)/\Gamma(p) $$ for $d\in\mathbb{R}$. If $p+d\le0$, then $EZ^d=\infty$. From (1) it follows that $$ EY^{d}ET^d=EZ^d\implies EY^d=\frac{\Gamma(p+d)}{\Gamma(p)}\bigg/\frac{\Gamma(p+q+d)}{\Gamma(p+q)} $$ where $p+d>0$ and $p+q+d>0$.