# Generating Function 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 $z^m G \left({z}\right)$ is the generating function for the sequence $\left\langle{a_{n - m} }\right\rangle$.

## Proof

 $\displaystyle z^m G \left({z}\right)$ $=$ $\displaystyle z^m \sum_{n \mathop \ge 0} a_n z^n$ Definition of Generating Function $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop \ge 0} a_n z^{n + m}$ $\displaystyle$ $=$ $\displaystyle \sum_{n + m \mathop \ge 0} a_{n - m} z^n$ Translation of Index Variable of Summation $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop \ge m} a_{n - m} z^n$

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

$z^m G \left({z}\right) = \displaystyle \sum_{n \mathop \ge 0} a_{n - m} z^n$

Hence the result.

$\blacksquare$