Weight-dependent commutation relations and combinatorial identities

We derive combinatorial identities for variables satisfying specific systems of commutation relations, in particular elliptic commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and $y$ satisfying $yx=qxy$. In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we derive results for an elliptic derivative of elliptic commuting variables, and finally study weight-dependent extensions of the Weyl algebra which we connect to rook theory.