In this Introduction to representation theory they define the '$q$-Weyl algebra by the primary defining relation $$xy = qyx$$
This seems appropriate in $q$-deformations based on the basic building block being a $q$-number: $${[n]}_q = \frac{1-q^n}{1-q}$$ Now if we were to extend to $(p,q)$-numbers: $${[n]}_{p,q} = \frac{p^n-q^n}{p-q}$$ Loosely speaking if we define an 'expression' in terms of $p$ and $q$ and denote it $f(p, q)$, what would a $(p,q)$-Weyl algebra's primary defining relation look like? $$xy = f(p,q)yx$$ And are there perhaps more than one of them? I'm not sure how to 'prove' or come to this solution myself without making any guesses as to what the central element $h$ defined by $xy - yx = h$ would be equal to. My guess is that the relation is $$xy = pqyx$$
Edit: I would also appreciate references if $(p,q)$-Weyl algebras have already been studied in detail. Thanks!
Edit 2: Is it perhaps $pxy = qyx$?