For z transform:

The continuous-time concept of time scaling does not directly extend to discrete time. However, the discrete time concept of time expansion i.e. of inserting a number of zeroes between successive values of a discrete time sequence can be defined. The new sequence can be defined as

x_(k)[n] = x[n / k] if n is a multiple of k

= 0 if n is not a multiple of k

has k - 1 zeroes inserted between successive values of the original sequence. This is known as upsampling by k. If

x[n] ↔ X(z) with ROC = R

 then x_(k)[n] ↔ X(z^k) with ROC = R^1/k i.e. z^k ∈ R

i.e. X(z^k) = S x[n](z^k)^-n, −∞ < n < ∞

= S x[n]z^-nk, −∞ < n < ∞