I am trying to find a (nontrivial) function $\mu$ satisfying the functional equation $$\mu(x,y)=\mu\bigg(xy,\frac{y-x}{xy}\bigg)$$ In case anybody wants to know, I plan on using $\mu$ to simplify and analyze the behaviors of the sequences $$\theta_{n+1}=\theta_n\phi_n,\space\space\space \phi_{n+1}=\frac{1}{\theta_n}-\frac{1}{\phi_n},\space\space\space \theta_0=\phi_0=1$$ Does anyone know how to find nontrivial $\mu$ satisfying the functional equation?
2026-03-26 16:07:00.1774541220
Finding an invariant $\mu(x,y)=\mu\big(xy,\frac{y-x}{xy}\big)$
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