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  "body": "\n## Definition\n\nLet $(x_n)_{n \\ge 0}$ be a sequence of complex numbers.\nIts ordinary [[generating function|0PHAC833JF]] is the power series $G(w) = \\sum_{n \\ge 0} x_n w^n$, with [[radius of convergence|0N288P2TB5]] $\\rho \\in [0, \\infty]$.\nIts [[unilateral Z-transform|EPG6JPVJE7]] is $X(z) = \\sum_{n \\ge 0} x_n z^{-n}$.\nThe *generating-function form of the Z-transform* is the identity\n$$X(z) = G(1/z) \\quad \\text{for } \\abs{z} \u003e 1/\\rho,$$\nwith the conventions $1/0 = \\infty$ and $1/\\infty = 0$.\n\n## Remarks\n\nExample: $x_n = a^n$ with $a \\ne 0$ has $G(w) = 1/(1 - a w)$ with $\\rho = 1/\\abs{a}$, and $G(1/z) = z/(z - a)$ for $\\abs{z} \u003e \\abs{a}$, which is its Z-transform.\nNon-example: a two-sided sequence with some $x_n \\ne 0$ for $n \u003c 0$ has a bilateral Z-transform with positive powers of $z$.\nThese become negative powers of $w$, so the result is a Laurent series in $w$ and not an ordinary generating function.\nUnder $w = 1/z$, the disc of convergence $\\abs{w} \u003c \\rho$ of $G$ becomes the outside $\\abs{z} \u003e 1/\\rho$ of a circle.\nThe shift rule $\\mathcal{Z}\\{(x_{n+1})\\} = z(X(z) - x_0)$ corresponds to $(G(w) - x_0)/w$ for generating functions.\n",
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