Where $X$ follows an F Ratio distribution F$(1,\alpha)$ with pdf: $$ f(x)= \frac{\alpha ^{\alpha /2} (\alpha +x)^{\frac{1}{2} (-\alpha -1)}}{\sqrt{x} B\left(\frac{1}{2},\frac{\alpha }{2}\right)},\; x\in [0,\infty).$$ Looking for the distribution of the $n$-summed independent F Ratio-distributed variables $Y= \sum_{1 \leq i \leq n}X_i$, with $\alpha>2$. I tried to work with the $n$-convoluted characteristic function: $$\chi_n(t)=\left( \frac{\Gamma \left(\frac{\alpha +1}{2}\right) U\left(\frac{1}{2},1-\frac{\alpha }{2},-i t \alpha \right)}{\Gamma \left(\frac{\alpha }{2}\right)}\right)^n$$ (where $U(.,.,.)$ is the confluent hypergeometric function with integral representation $U(a,b,z)=\frac{1}{a \Gamma }\int _0^{\infty } t^{a-1} (t+1)^{-a+b-1} e^{t (-z)} \mathrm{d} t$ ) and was unable to go anywhere. I can pull the moments from $\chi(t)$ (which turn out to be rapidly infinite at higher orders) but I am interested in the density.
With gratitude.