Difference of two Gamma random variables Let $X$ and $Y$ be two i.i.d Gamma random variables with a integer parameter $\alpha$, i.e, $X \sim \Gamma(\alpha,1)$. 
I wish to find the density of $X-Y$, however the integrals are turning out be messy. Can anyone help me in this? 
My actual goal is to compute the asymptotics of the entropy of $X-Y$ as $\alpha$ becomes large, where $h(Z)=\int f(z) \log \frac{1}{f(z)} dz$ is the entropy of any random variable with density $f$.
 A: When the shape parameter $\alpha$ is large and the rate/scale parameter $\beta = 1$ is fixed, the distributions of $X$ and $Y$ are approximately normal with mean and variance both equal to $\alpha$.  Therefore, their difference $W = X-Y$ is approximately normal with mean $0$ and variance $2\alpha$, with PDF $$f_W(w) \approx \frac{e^{-w^2/(4\alpha)}}{\sqrt{4\pi\alpha}}.$$  Then $$\operatorname{E}[-\log f(W)] \approx \operatorname{E}\left[\frac{W^2}{4\alpha} + \frac{1}{2} \log 4\pi \alpha\right] = \frac{1}{2}\log 4 \pi \alpha + \frac{1}{4\alpha}(\operatorname{Var}[W] + \operatorname{E}[W]^2) = \frac{1}{2}(1 + \log 4\pi \alpha),$$ which is the asymptotic behavior of the entropy for large $\alpha$.
A: An answer to the question about the density of the difference of two Gamma random variables can be found in this answer of mine on stats.SE.  As noted
there, in the general case, the integrals are not straightforward to evaluate. In the special case above when
the random variables are i.i.d. and the order parameter $\alpha$ is an
integer as in the OP's case, the integrals are still not easy but
have been studied before. Related integrals are
listed in Gradshteyn and Ryzhik, Tables of Integrals, Series, and Products, Section 3.383, and the density can be determined from
these.
