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

 $\ds z^{-m} \map G z$ $=$ $\ds z^{-m} \sum_{n \mathop \ge 0} a_n z^n$ Definition of Generating Function $\ds$ $=$ $\ds \sum_{n \mathop \ge 0} a_n z^{n - m}$ $\ds$ $=$ $\ds \sum_{n + m \mathop \ge 0} a_{n + m} z^n$ Translation of Index Variable of Summation $\ds$ $=$ $\ds \sum_{n \mathop \ge 0} a_{n + m} z^n + \sum_{k \mathop = -m}^{-1} a_{k + m} z^k$ splitting up and changing variable $\ds$ $=$ $\ds \sum_{n \mathop \ge 0} a_{n + m} z^n + \sum_{k \mathop = 0}^{m - 1} a_k z^{k - m}$ Translation of Index Variable of Summation $\ds$ $=$ $\ds \sum_{n \mathop \ge 0} a_{n + m} z^n + z^{-m} \sum_{k \mathop = 0}^{m - 1} a_k z^k$

Hence the result.

$\blacksquare$