# Computations on U and V when they do not commute.

We have $f= U + U^{-1} + V + V^{-1}$. Exp is the exponential map: $e^{f}=\Sigma_{m=0}^{\infty} \frac{f^{n}}{n!}$. U and V do not commute and we have $U^{m}V^{n}= e^{imn}V^{n}U^{m}$ for any m,n integer. I want to find the constant term for the expression $\partial_{2}e^{-f/2} \partial_{2}e^{f}$. We have $\partial_{2}f= V - V^{-1}$.

Is there any program in which I can compute it?

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What is (mathematically) $\partial_2$? Is it a derivation? (So that we can formally extend it to a polynomial to a series in $f$...) Is there really $\exp(imn)$ the twisting factor?