Generating Function 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 $z^m G \left({z}\right)$ 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.

$\blacksquare$


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