Let $A$, $B$ and $C$ be some operators (details not relevant) and $\alpha$ a constant. The operators obey the following commutation relations:
$$ [A,B]=\alpha$$ $$ [C,B]=2A$$ $$ [A,C]=0$$
I want to evaluate the following product,
$$ \prod\limits_{k=1}^{n} \left( 1 - \frac{BA}{\alpha k} + \frac{B^2 C}{2 k \alpha (2k - 1)} \right) $$
but I have no idea how to proceed and I've been thinking about it for days. All help would be greatly appreciated!