Z-transform property proof

We use y(k) = Σ_{m=0}^{k} x(m) (the running sum of x).

Proof. Write the difference of consecutive terms, with y(−1) = 0:

y(k) − y(k − 1) = x(k)

Take the Z-transform of both sides. By the right-shift property, Z[y(k − 1)] = z⁻¹ Y(z) (since y(−1) = 0):

Y(z) − z⁻¹ Y(z) = X(z)
Y(z) (1 − 1/z) = X(z)
Y(z) = X(z) / (1 − z⁻¹) = (z/(z − 1)) X(z)

Alternative (direct) proof: Y(z) = Σₖ z^(−k) Σ_{m≤k} x(m). Interchanging the sums,

Y(z) = Σ_m x(m) Σ_{k≥m} z^(−k) = Σ_m x(m) z^(−m) · 1/(1 − z⁻¹) = X(z) · z/(z − 1)

for |z| > 1. ∎