  Given a weighted $\ell^2$ space with weights associated to an entire
function, we consider pairs of weighted shift operators, whose commutators are
diagonal operators, when considered as operators over a general Fock space. We
establish a calculus for the algebra of these commutators and apply it to the
general case of Gelfond-Leontiev derivatives. This general class of operators
includes many known examples, like classic fractional derivatives and Dunkl
operators. This allows us to establish a general framework which goes beyond
the classic Weyl-Heisenberg algebra. Concrete examples for its application are
provided.
