Laplace Transform of Generating Function of Sequence

Theorem
Let $\left\langle{a_n}\right\rangle$ be a sequence which has a generating function which is convergent.

Let $G \left({z}\right)$ be the generating function for $\left\langle{a_n}\right\rangle$.

Let $f \left({x}\right)$ be the step function:
 * $f \left({x}\right) = \displaystyle \sum_{k \mathop \in \Z} a_k \left[{0 \le k \le x}\right]$

where $\left[{0 \le k \le x}\right]$ is Iverson's convention.

Then the Laplace transform of $f \left({x}\right)$ is given by:
 * $\mathcal L \left\{ {f \left({s}\right)}\right\} = \dfrac {G \left({e^{-s} }\right)} s$

Proof
We note that:

and:

Then: