Product of Exponential Generating Functions

Theorem
Let $\map G z$ be the exponential generating function for the sequence $\sequence {\dfrac {a_n} {n!} }$.

Let $\map H z$ be the exponential generating function for the sequence $\sequence {\dfrac {b_n} {n!} }$.

Then $\map G z \map H z$ is the generating function for the sequence $\sequence {\dfrac {c_n} {n!} }$, where:


 * $\forall n \in \Z_{\ge 0}: c_n = \ds \sum_{k \mathop \in \Z} \dbinom n k a_k b_{n - k}$