  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.
