I doubt that this will get you a closed form but did you try using characteristic functions?
Let where and using this parametrization of the Gamma distribution. Also assume and are independent. Then what is the distribution (pdf) of ?
This question doesn't seem as straightforward as it sounds. For example, I have tried using the convolution formulas here, but can't seem to find a closed form expression for the integral. I have also tried multiplying the moment generating functions (mgfs) of and , but it does not seem to match up to any known mgfs.
Does anyone have any ideas on how to find the distribution for ?
it's basically the fourier transform of a probability distribution function and as such the characteristic function of the sum of 2 rvs is the product of the individual characteristic functions.
I just came across this
it won't give you a closed form answer but it might be of interest.