The MGF of $X_{(1)}$ when $X_1,X_2$ are i.i.d Gamma variables

Question

I want to find the moment generating function (MGF) of the order statistic $X_{(1)}$ where $X_1, X_2$ are iid $\Gamma(\alpha, 1)$.

First attempt was to simply find the order statistic pdf in the usual way, followed by finding the the MGF in the usual way. However, things got tricky when calculating the CDF for the gamma distribution. I think there is a trick that I might be missing.

EDIT:

This is the pdf for the version of the gamma distribution I am using:

$$f(x) = \frac{\beta^\alpha}{\Gamma (\alpha)} x^{\alpha -1} e^{-\beta x}$$

Answer 1

The CDF for your $\Gamma(\alpha,1)$ distribution can be written in terms of the Whittaker M function:

$$F(x) = \frac{x^{\alpha/2} e^{-x/2}}{\Gamma(\alpha+2)} M(\alpha/2,(1+\alpha)/2,x) + \frac{x^\alpha e^{-x} (1+\alpha)}{\Gamma(\alpha+2)}$$