Convolution of mixture of Erlangs

Let $P'(x)=\sum_{j=1}^n a_j\frac{a^jx^{j-1}}{(j-1)!}e^{-ax}$ be the density function of a mixture of Erlangs and let $\alpha\in(0,1)$. Is is possible to determine an analytic expression for $\sum_{n=1}^\infty (1-\alpha) \alpha^nP^{*n}(x)$, where $P^{*n}$ is the $n-$fold convolution of $P$ with itself? I was thinking about calculating $\mathcal L^{-1}([\mathcal L(P)]^n)$, where $\mathcal L$ is the Laplace transform, but this looks really hard.