I want to know how to simplify the following expression by using the fact that $\sum_{i=0}^\infty \frac{X^i}{i!}=e^X$. The expression to be simplified is as follows:
$\sum_{i=0}^{\infty} \sum_{j=0}^i \frac{X^{i-j}}{(i-j)!} \cdot \frac{Y^j}{j!}\;,$ where $X$ and $Y$ are square matrices (not commutative). (That is, $X\cdot Y \neq Y \cdot X$).