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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!

lel
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  • If I were you, I would look for a finite $n$ dimensional realization (with $n$ small, if it is possible) I mean with operators replaced by matrices and gain understanding of the problem by making explicit computations and trying to find structures.. 2) Could you say something about your motivation for this rather particular problem ?
    • – Jean Marie Nov 01 '17 at 22:02
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    @JeanMarie: there are no finite dimensional realizations (just take the trace of the equation $[A,B] =\alpha I $ on both sides, see here) – Fabian Nov 01 '17 at 22:13
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  • I explored it explicitly for lower values of $n$, it yields $\frac{(n+2)(n+1)}{2}$ distinct terms but I wasn't able to extract a pattern.
  • It is a distillation of a previous question, stripped down to its bones, because the last question was left unanswered (https://math.stackexchange.com/questions/2476156/how-to-solve-a-recursion-relation-on-tensors-including-derivatives-and-traces)
    • – lel Nov 01 '17 at 22:14
  • @Fabian My bad. You are perfectly right. I should have taken care. – Jean Marie Nov 01 '17 at 22:37