Distribution of sum of normal and gamma random variable

#1
Let [math]Z = X + Y[/math] where [math]X \sim N\left(\mu, \sigma^2 \right)[/math] and [math]Y \sim \Gamma\left(k, \theta \right)[/math] using this parametrization of the Gamma distribution. Also assume [math]X[/math] and [math]Y[/math] are independent. Then what is the distribution (pdf) of [math]Z[/math]?


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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 [math]X[/math] and [math]Y[/math], but it does not seem to match up to any known mgfs.

Does anyone have any ideas on how to find the distribution for [math]Z[/math]?