Are Z-transform time shifting and differentiation properties always compatible?

1. What is the time shifting property of Z-transform?
2. What is differentiation in z domain property of Z-transform?

What is the time shifting property of Z-transform?

Time Shifting: If we have a time-shifted sequence such as x[n-k], then its z-transform is given by Z x[n-k] = z^-kX(z). Let's take n – k = m, i.e., n = k + m and y[n] = x[n-k].

What is differentiation in z domain property of Z-transform?

A well-known property of the Z transform is the differentiation in z-domain property, which states that if X(z) ≡ Zx[n] is the Z transform of a sequence x[n] then the Z transform of the sequence nx[n] is Znx[n]=−z(dX (z)/dz).