Let where and using this parametrization of the Gamma distribution. Also assume and are independent. Then what is the distribution (pdf) of ?

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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 ?