Generating Function of Multiple of Parameter

Theorem

Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$.

Let $c$ be a constant.

Then $G \left({c z}\right)$ be the generating function for the sequence $\left\langle{b_n}\right\rangle$ where:

$\forall n \in \Z_{\ge 0}: b_n = c^n a_n$

$\ds G \left({c z}\right)$

$=$

$\ds \sum_{n \mathop \ge 0} a_n \left({c z}\right)^n$

$\ds $

$=$

$\ds \sum_{n \mathop \ge 0} \left({a_n c^n}\right) z^n$

Hence the result by definition of generating function.

$\blacksquare$