Product of Exponential Generating Functions

Theorem
Let $G \left({z}\right)$ be the exponential generating function for the sequence $\left\langle{\dfrac {a_n} {n!} }\right\rangle$.

Let $H \left({z}\right)$ be the exponential generating function for the sequence $\left\langle{\dfrac {b_n} {n!} }\right\rangle$.

Then $G \left({z}\right) H \left({z}\right)$ is the generating function for the sequence $\left\langle{\dfrac {c_n} {n!} }\right\rangle$, where:


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