Showing an equality using a dynamical system

Question

I'm asked to show that $\sum_{n=0}^{+\infty} \frac{(x+y)^n}{n!} =\sum_{n=0}^{+\infty} \frac{x^n}{n!}\sum_{n=0}^{+\infty} \frac{y^n}{n!}$ using a dynamical system. However, I couldn't figure out how to start. I thought that $$ x' = x \\ x(0)=1$$ can be considered. And, the fact that time maps constitute a group under composition can be useful. However, I need some hint to continue. Thanks in advance.

Answer 1

The solutions of $x'=x$ with $x(0)=x_0$ are $\phi_t(x_0)=e^tx_0$ ($x(0)=1$ is not sufficient for what you want). So it follows from $$ \phi_{x+y}=\phi_x\circ\phi_y $$ that $$ e^{x+y}=e^{x}e^{y}. $$

Answer 2

I suppose that $x$ and $y$ are scalars (f there are matrix, they must commute in general).

The trick is to use the uniqueness in Cauchy Lipschitz theorem (wich gives the group property of the flow of an ODE).

Let $z$ be the solution of $z'(t) = (x+y) z(t)$ with the initial condition $z(0)=1$. This is a linear ODE, and we know that the solution is unique. But $t \to \exp((x+y) t)$ and $t \to \exp(x t) \exp(y t)$ are solutions. This gives the result.

I think that this argument cannot work with matrix because it is circular. Indeed to prove that $(\exp(t A)) '= \exp(A t) A$ we need to prove that $\exp(A+B) =\exp(A) + \exp(B)$ if $A$ and $B$ commute.