Generating Function Divided by Power of Parameter

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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.

$\blacksquare$


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