Z-transform as a generating function

Summary

The unilateral Z-transform of a sequence is its ordinary generating function evaluated at $1/z$. So the two notions carry the same information and obey the same algebra, with the variable inverted. Engineers use the Z-transform in the variable $z$, and combinatorialists use generating functions in $w = 1/z$. Results proved on one side, such as rules for shifts, convolution and partial fractions, transfer to the other at once.