I have come across a generating function that is similar to another generating function.
First, some preliminaries. I call a closed form a function like $\frac{1}{1-x}$. In other words, a closed form does not involve a summation or integral, but rather a bunch of arithmetic that describes a generating function.
Now, I have come across a closed form of the generating function:
$$\sum_{k=0}^\infty{X_k y^k} = X_0 + X_1 y^1 + X_2 y^2 + \dots$$
where the $X_k$'s are themselves generating functions:
$$X_k = \frac{1}{1-(z_k)} = 1 + (z_k) + (z_k)^2 + \dots$$
I'd like to convert the $X_k$'s into $z_k$'s. In other words, each coefficient of the original series is $\frac{1}{1-(z_k)}$. I'd like to convert from that to just $z_k$. I'm wondering if there is some sort of transformation that can do this all at once.
I would greatly appreciate any help on this.
IMPORTANT EXPLANATION
I know how to get the $z$ values in the formula. What I really want is to change the formula into something that is not equal to the original formula. In other words, say I have:
$$y\frac{1}{1-z_1}+y^2\frac{1}{1-z_2}$$
Then I want:
$$y z_1 + y^2 z_2$$