Generating Function 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 $z^m \map G z$ is the generating function for the sequence $\sequence {a_{n - m} }$.

Proof
By letting $a_n = 0$ for all $n < 0$:


 * $z^m \map G z = \ds \sum_{n \mathop \ge 0} a_{n - m} z^n$

Hence the result.