probability generating function of a poisson random variable

i should start by asking is anyone on the forum well versed in the coalescent theory?

i have a problem in a derivation

if the number of mutations is given by

as one can see this is the probability generating function of a poisson random variable with random mean. where

now from here i can write this as a product

so

Now im confused what to do. If i have the probability generating function how do i find the density function (derivative?)

the next step in the derivation should be

I cant for the life of me figure out how this was found from the pgf. PLEASE ANY IDEAS ARE WELCOME, SO IF NO ONE KNOWS THE ANSWER SUGGEST ANYTHING.

many thanks

chogo