probability - Poisson distribution with Poisson parameter

I have a problem with following exercise (it comes from Geoffrey G. Grimmett, David R. Stirzaker, Probability and Random Processes, Oxford University Press 2001, page 161, ex. 3a):

Let $X$ have the Poisson distribution with parameter $Y$ where $Y$ has the Poisson distribution with parameter $\mu$. Show that

$$G_{X+Y}(s) = e^{\mu(s e^{s-1} - 1)}$$

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So $G_Y(s) = e^{\mu(s-1)}$
Now I want to compute $G_X$ (is this approach correct?)

$$G_X(s) = \sum_{x=0}^{\infty} s^x P(X=x) = \sum_{x=0}\sum_{y=0} s^x \frac{e^{-y} y^x}{x!} \frac{e^{-\mu} \mu^y}{y!}$$
$$G_X(s) = e^{-\mu} \sum_{x=0} \frac{s^x}{x!} \sum_{y=0} \frac{y^x e^{-y}\mu^y}{y!}$$

And I don't know how to cope with this summation.

Secondly, are these variables independent? I.e. can I use then following formula?

$$G_{X+Y}(s) = G_X(s) G_Y(s)$$

Thanks for your help.