I was reading about the matrix exponential function and I came across this:

If $xy = yx$ then

$$ \exp(x+y) = \exp(x)\cdot\exp(y) $$

My textbook gives a proof as follows:

$$ \exp(x+y) = \sum\limits_{k=0}^{\infty}\frac{1}{k!}(x+y)^{k} = \sum\limits_{k=0}^{\infty}\left(\sum\limits_{l=0}^{k}\frac{x^{l}y^{k-l}}{l!(k-l)!}\right) = \left(\sum\limits_{p=0}^{\infty}\frac{x^{p}}{p!}\right)\cdot\left(\sum\limits_{l=0}^{\infty}\frac{y^{l}}{l!}\right)$$

I have trouble understanding the last equality. I guess it has something to do with Fubini but I do not understand how an infinite and finite summation got changed into two infinite summations.

Any help will be appreciated.

EDIT: The sole comment was enough to get me to the answer. This is nothing but the convolution product and the result follows from Merten's Theorem.


So that this question doesn't remain in the unanswered queue:

We note that $$ \sum\limits_{k=0}^{\infty}\left(\sum\limits_{l=0}^{k}\frac{x^{l}y^{k-l}}{l!(k-l)!}\right) $$ Is simply the Cauchy Product (aka the convolution product) of the two series $ \sum_{p=0}^{\infty}\frac{x^{p}}{p!} $ and $ \sum_{l=0}^{\infty}\frac{y^{l}}{l!} $ . By Merten's Theorem, we can deduce that $$ \sum\limits_{k=0}^{\infty}\left(\sum\limits_{l=0}^{k}\frac{x^{l}y^{k-l}}{l!(k-l)!}\right) = \left(\sum\limits_{p=0}^{\infty}\frac{x^{p}}{p!}\right)\cdot\left(\sum\limits_{l=0}^{\infty}\frac{y^{l}}{l!}\right) $$


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