# Solved – Inverting the Fourier Transform for a Fisher distribution

The characteristic function of Fisher \$mathcal{F}(1,alpha)\$ distribution is:
\$\$C(t)=frac{Gamma left(frac{alpha +1}{2}right) Uleft(frac{1}{2},1-frac{alpha }{2},-i t alpha right)}{Gamma left(frac{alpha }{2}right)}\$\$
where \$U\$ is the confluent hypergeometric function. I am trying to solve the inverse Fourier transform \$mathcal {F} _ {t,x}^{-1}\$ of the \$n\$-convolution to recover the density of a variable \$x\$, that is:
\$\$mathcal {F} _ {t,x}^{-1} left(C(t)^nright)\$\$
with the purpose of getting the distribution of the sum of \$n\$ Fisher-distributed random variables. I wonder if someone has any idea as it seems to be very difficult to solve. I tried values of \$alpha=3\$ and \$n=2\$ to no avail.
Note: for \$n=2\$ by convolution I get the pdf of the average (not sum):

\$\$frac{3 left(12 left(x^2+3right) left(5 x^2-3right) x^2+9 left(20 x^4+27 x^2+9right) log left(frac{4 x^2}{3}+1right)+2 sqrt{3} left(x^2+15right) left(4 x^2+3right) x^3 tan ^{-1}left(frac{2 x}{sqrt{3}}right)right)}{pi ^2 x^3 left(x^2+3right)^3 left(4 x^2+3right)}\$\$,

where \$x\$ is an average of 2 variables. I know it is unwieldy but would love to get an idea of the approximation of the basin distribution.

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