Z-transform as a generating function

Remarks

Example: $x_n = a^n$ with $a \ne 0$ has $G(w) = 1/(1 - a w)$ with $\rho = 1/\abs{a}$, and $G(1/z) = z/(z - a)$ for $\abs{z} > \abs{a}$, which is its Z-transform. Non-example: a two-sided sequence with some $x_n \ne 0$ for $n < 0$ has a bilateral Z-transform with positive powers of $z$. These become negative powers of $w$, so the result is a Laurent series in $w$ and not an ordinary generating function. Under $w = 1/z$, the disc of convergence $\abs{w} < \rho$ of $G$ becomes the outside $\abs{z} > 1/\rho$ of a circle. The shift rule $\mathcal{Z}\{(x_{n+1})\} = z(X(z) - x_0)$ corresponds to $(G(w) - x_0)/w$ for generating functions.