Advanced Z-transform

Remarks

For $m = 0$ it is the ordinary Z-transform of the samples $f(kT)$. Example: for $f(t) = e^{-at}$ with $a \in \C$, $F(z, m) = e^{-am} \sum_{k \ge 0} (e^{-aT} z^{-1})^k = e^{-am} \, z/(z - e^{-aT})$ for $\abs{z} > \abs{e^{-aT}}$. For $f(t) = t$, $F(z, m) = \sum_{k \ge 0} (kT + m) z^{-k} = Tz/(z - 1)^2 + m z/(z - 1)$ for $\abs{z} > 1$. Non-example: the sum $\sum_k f(k T + m) z^{-k}$ with $m$ outside $[0, T]$ is not covered; for $m = T$ it is not a new transform either, since $F(z, T) = z (F(z, 0) - f(0))$. Some texts call a closely related transform, with a delay parameter in place of the advance, the modified Z-transform.