Consider $\mathbb C \langle x, y \rangle / \langle x y - \lambda y x - \mu \rangle$ for some constants $\lambda, \mu \in \mathbb C$. I am trying to obtain a closed formula for $x^m y^n$, for general $m, n \in \mathbb N$, in terms of monomials of the form $y^k x^l$.
We have $x y^n = \lambda^n y^n x + \mu (1 + \lambda + \dotsb + \lambda^{n-1}) y^{n-1}$ and since $x^m y^n = x^{m-1} (x y^n)$ we thus get $$ x^m y^n = x^{m-1} y^n x \lambda^n + x^{m-1} y^{n-1} \mu (1 + \lambda + \dotsb + \lambda^{n-1}) $$ which looks like a recurrence relation, but I don't know any techniques for writing down a closed formula of the form $$ x^m y^n = \sum a_{kl} y^k x^l. $$
It looks like the coefficients of $y^k x^l$ should be related to some kind of binomial coefficients for a "weighted" Pascal's triangle, where the usual Pascal's triangle has weights $1$ and the binomial coefficient is just counting the numbers of paths from the top.
