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  "body": "\n## Summary\n\nThe [[unilateral Z-transform|EPG6JPVJE7]] of a sequence is its ordinary [[generating function|0PHAC833JF]] evaluated at $1/z$.\nSo the two notions carry the same information and obey the same algebra, with the variable inverted.\nEngineers use the Z-transform in the variable $z$, and combinatorialists use generating functions in $w = 1/z$.\nResults proved on one side, such as rules for shifts, convolution and partial fractions, transfer to the other at once.\n\n## Explanations\n\n### L1 human\n\nA generating function packs a sequence $x_0, x_1, \\ldots$ into $x_0 + x_1 w + x_2 w^2 + \\cdots$.\nThe Z-transform packs it into $x_0 + x_1 z^{-1} + x_2 z^{-2} + \\cdots$.\nThese are the same series with $w = 1/z$.\nFor the constant sequence $1, 1, 1, \\ldots$, the generating function is $1/(1 - w)$ and the Z-transform is $z/(z - 1)$.\n",
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