I calculated the Moment Generating Function of a Poisson random variable: $N \sim \mathrm{Poiss}(\lambda)$ $$ E[e^{t N}] = \sum\limits_{k=0}^\infty e^{t k} \frac{e^{-\lambda}\lambda^k }{k!} = e^ {\lambda[e^{t}-1]} $$
I also calculated the Generating Function of $N$: $$ E[t^{N}] = e^ {\lambda[{t}-1]} $$
Now, I want to verify the relationship between the Generating Function (GF) and the Moment Generating Function (MGF).
Specifically, I want to check that GF = MGF $\cdot \log t$.
In other words, GF / MGF should be equal to $\log t$.
But when I work it out, I am stuck at the below point:
The GF divided by the MGF gives me:
GF / MGF = $$ e^ {\lambda[t - e^{t}]} $$
I tried various ways to see if this reduces to $\log t$, but my methods didn't seem to proceed. Any advice or insights would be very helpful here. Thanks a lot.