Lets consider the following symmetrical complex relation
$f(z) = \frac{B(z)B(\frac{1}{z})}{B(z)B(\frac{1}{z}) + \alpha A(z)A(\frac{1}{z})}$
where $\alpha$ is a real value, $B(z)=\sum_{i=1}^{N}\beta_i z^i$ and $A(z)=\sum_{i=1}^{M}\alpha_i z^i$. I am looking for a function $G(z)$ (as a function of $A$ and $B$) the satisfy the following relation
$f(z) = G(z)G(\frac{1}{z})$