Generating Function Divided by Power of Parameter

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

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

Then $\dfrac 1 {z^m} \paren {\map G z - \ds \sum_{k \mathop = 0}^{m - 1} a_k z^k}$ is the generating function for the sequence $\sequence {a_{n + m} }$.

Proof
Hence the result.