Generating Function Divided by Power of Parameter

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

Let $m \in \Z_{\ge 0}$ be a non-negative integer.

Then $\dfrac 1 {z^m} \left({G \left({z}\right) - \displaystyle \sum_{k \mathop = 0}^{m - 1} a_k z^k}\right)$ is the generating function for the sequence $\left\langle{a_{n + m} }\right\rangle$.

Proof
Hence the result.