Shift property of the Z-transform

Remarks

Each clause is a change of summation index. For clause 1, $\sum_n x_{n-k} z^{-n} = z^{-k} \sum_m x_m z^{-m}$ with $m = n - k$, and the same computation with absolute values shows that the regions agree. For clause 3, $\sum_{n \ge 0} x_{n+k} z^{-n} = z^{k} \sum_{m \ge k} x_m z^{-m}$, and the sum over $m \ge k$ is $X(z)$ minus its first $k$ terms. Example: for $x_n = 1$ ($n \ge 0$), $X(z) = z/(z-1)$, and clause 3 with $k = 1$ gives $z (z/(z-1) - 1) = z/(z-1)$, as it must, since the shifted sequence is again constant. Clause 2 needs $x_n = 0$ for $n < 0$; otherwise the terms $x_{-k}, \ldots, x_{-1}$ also enter the unilateral transform of the delayed sequence.

Going deeper

For a student

Clause 3 is the discrete analogue of $\mathcal{L}\{f'\}(s) = s F(s) - f(0)$ for the Laplace transform. It is what makes the Z-transform solve linear difference equations with constant coefficients: each shifted unknown becomes $z^k$ times the transform, plus a polynomial built from the initial values. Clause 1 shows that a time-invariant system acts on transforms by multiplication.