Let $Z = X + Y$ where $X \sim N\left(\mu, \sigma^2 \right)$ and $Y \sim \Gamma\left(k, \theta \right)$ using this parametrization of the Gamma distribution. Also assume $X$ and $Y$ are independent. Then what is the distribution (pdf) of $Z$?
I thought this question looked straightforward, but it doesn't seem so. 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 $X$ and $Y$, but it does not seem to match up to any known mgfs.
Is it possible to find a closed-form solution for the distribution of $Z$? If not, how about some sort of approximation?