I'm studying generating functions. I'm trying to understand exercise 6.7 in this article: https://www.gameludere.it/2019/05/19/funzioni-generatrici-esponenziali-e-applicazioni/#1_Le_funzioni_generatrici_esponenziali
But there is a step I don't understand.
That is, he has this generating function: $$G_{r}(x)=\sum_{n=0}^{\infty}{n \brace r} \frac{x^{n}}{n!}$$ and use $G_{r}(x)=G_{1}(x)\cdot G_{r-1}(x)$ to prove that $$G_{r}(x) = \frac{(e^x-1)^r}{r!}$$
How does he know that $G_{r}(x)=G_{1}(x)\cdot G_{r-1}(x)$?
I've tried to do second member manipulation, but I don't find any simple method to verify the identity. I suppose there is some smart trick to apply.