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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.

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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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