Suppose we have the element $a$ and $b$ in some algebra $A$ and $0<q<1$ subject to the relations:
$$a^2b-(q+q^{-1})aba+ba^2=0$$ $$b^2a-(q+q^{-1})bab+ab^2=0$$
I want to deduce from this a closed formula for $b(ba)^n$. I think you have to use an inductive argument, but I don't see it. Anybody who can help me?