I need help finishing with the derivation of a formula. I am not sure on how to link the derivation from one step to the final derivation.
From here, I am somehow supposed to get
Just for my convenience, I set $h \equiv z$ and $z \equiv \zeta$.
This is where I am stuck and I need a little help. It seems that I am getting closer to that final step, but why and how do $e^{-2i\omega(x-y_j)}$ and $e^{2i\omega(x-y_{j-1})}$ change so that I have $\bar{r}_j$ and $z_j$, respectively?
As you can see, it has something to do with a recursive algorithm, but it also mentioned about disk automorphism and Fourier transform in which I have no or little knowledge.
Please help and also explain how you reach the final formula. Thank you.