Problem: Find f(k), when F(z) = (z^2)/[(z-1)(z+1)(z-0.5)]

Step: 1.) Partial fraction expansion (no problems here).

F(z) = 1/(z-1) + (1/3)/(z+1) - (1/3)/(z-0.5)

Step: 2.) Inverse transform back using table properties (my problem)

I am having trouble inverse transforming back to the discrete function.

Matlab gives the answer:f(k) = 1 - (2/3)(1/2)^k - (1/3)(-1)^k

This answer seems to arise if I multiply both sides of the partial expansion by z.

Which would yield:

z F(z) = z/(z-1) + 1/3z/(z+1) - 1/3z/(z-0.5)

The right side of the equation agrees with matlab after inverse transforming.

However this would give the answer of f(k+1) while I am trying to find f(k). This is due to the z property f(k-L) <--> (z^-L) * F(z) :

f(k+1) = 1 - (2/3)(1/2)^k - (1/3)(-1)^k

It seems as if f(k+1) is analyzed as the same thing as f(k).

I think I am having a lapse in regards to relating f(k) and f(k+1).

Could someone help me understand this?