Generating Function Divided by Power of Parameter

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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.

$\blacksquare$


Sources

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