# 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

 $\displaystyle z^{-m} G \left({z}\right)$ $=$ $\displaystyle z^{-m} \sum_{n \mathop \ge 0} a_n z^n$ 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 $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop \ge 0} a_{n + m} z^n + \sum_{k \mathop = -m}^{-1} a_{k + m} z^k$ splitting up and changing variable $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop \ge 0} a_{n + m} z^n + \sum_{k \mathop = 0}^{m - 1} a_k z^{k - m}$ Translation of Index Variable of Summation $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop \ge 0} a_{n + m} z^n + z^{-m} \sum_{k \mathop = 0}^{m - 1} a_k z^k$

Hence the result.

$\blacksquare$