A discrepancy in understanding another formula.

Question

If I have:

$$\phi_{n}(z) = \chi_{n}(A(z)),(z \in \mathbb{C}, |z| = 1) $$ and $$\chi_n(A(z))=z^n+z^{n-1}z^{-1}+\cdots zz^{-n+1}+z^{-n},$$

Why $z^m - z^{-m} = \phi_{m} - \phi_{m-1}$?

Answer 1

This is not true. Just look at a few examples: $$\phi_1 = z + z^{-1}$$ $$\phi_2 = z^2 + 1 + z^{-2}$$ $$\phi_3 = z^3 + z + z^{-1} + z^{-3}$$ $$\phi_4 = z^4 + z^2 + 1 + z^{-2} + z^{-4}.$$ What instead seems to be true is that $$\phi_n - \phi_{n-2} = z^n + z^{-n}$$ which you can prove using $$\phi_n(z) = \frac{z^{n+1} - 1/z^{n+1}}{z - 1/z}.$$