How can one solve this functional equation $f(u)f(v)=f(u+v)(f(u)+f(v)-(q+q^{-1}))+1$?

Question

Solve for $f$: $$f(u)f(v)=f(u+v)(f(u)+f(v)-(q+q^{-1}))+1$$ where $q$ is a fixed complex value and $f$ is defined on $\mathbb R$.

We can assume $f(0)=q$ by symmetry. What I got is $f$ takes values in $\{q,q^{-1}\}$ and two relations $$f(u)=f(v)=q\Rightarrow f(u+v)=q,\ f(u+v)=f(v)^{-1}=q\Rightarrow f(u)=q.$$ So for example $$f(x)=\begin{cases}q\quad x\le 0\\q^{-1}\quad x>0\end{cases}$$ can be a solution. Also by those two relations if $f(x_0)$ is known then $f(kx_0)$ is known for $k\in\mathbb Q$.

Does this equation only give trivial solutions such as the one I have posted? Or is there any "horrible" solution?

Answer 1

It has some wild solutions. You can decompose $\Bbb R$ into two "wild" additively closed sets $X_1$ and $X_2$. Additively closed means that for $x,y\in X_i$ we also have $x+y\in X_i$. Define

$$f(x)=\begin{cases} q & \text{for }x\in X_1\\ q^{-1} & \text{for }x\in X_2\\ \end{cases}.$$

Your functional equation only describes dependencies between values from the same set.

Answer 2

E.g. $v:=0$ : $\,\displaystyle f(u)f(0)=f(u)(f(u)+f(0)-(q+q^{-1}))+1 $

It follows $\,\displaystyle f(u)^2-(q+q^{-1})f(u)+1=(f(u)-q)(f(u)-q^{-1})=0 $

and therefore: $f(u)\in\{q,q^{-1}\}$

Note:

If $f(u)$ is real and therefore $\,q+q^{-1}=f(u)+f(u)^{-1}\,$ is real then it means that $\,|q|=1\,$ .