I'm trying to derive the overall Z-transform for the K-path sampling system shown below. I numbered some of the equations that I'll reference.

I don't think I understand going from equation 3 to equation 4. I understand that $\frac{V_{o,i}(z)}{V_{in,i}(z)} = H(z)$, but why is $\frac{V_{o,i}(z^{1/k})}{V_{in,i}(z^{1/k})} = H(z)$? Does this follow from equation 1?

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