I am trying to follow the following proof from Gut's book An Intermediate Course in Probability.
Let $X$ and $N$ be random variables.
$N \sim Po(\lambda)$ and $X|N=n \sim\operatorname{Bin}(n,p)$ the probability generating function is defined as $E[t^{X}]$ Find the probability generating function for X.
The steps taken in the book are the following. $g_{X}(t) = E[t^{X}] = E[E[t^{X}|N=n]] = E[(q+pt)^{n}]$ I get this step. Definition of probability generating function + law of iterated expectation. The next step is where I get stuck.
$g_{X}(t) = E[(q+pt)^{n}] = g_{N}(q+pt) = e^{\lambda((q+pt)-1)}$ in particular what confuses me is the step $E[(q+pt)^{n}] = g_{N}(q+pt)$
The only thing I can think of that reminds me of this is the convolution/multiplication duality of the transform. That this would be the PGF of an N-fold convolution of i.i.d random variables.
I would prefer your earlier step to be written like
$$g_{X}^{\,}(t) = E_{X}^{\,}[t^{X}]= E_N^{\,}[E_{X}^{\,}[t^{X}\mid N=n]] = E_N^{\,}[(q+pt)^{N}]$$
and, since the generating function for $N$ is $g_{N}^{\,}(s) =E_N^{\,}[s^{N}]$, you can let $s=q+pt$ and thus $$g_{N}^{\,}(q+pt) =E_N^{\,}[(q+pt)^{N}]$$
implying $$g_{X}^{\,}(t) = g_{N}^{\,}(q+pt)$$