We present a new method for determining the renormalization of Green functions to all orders in perturbation theory, which we call the displacement operator formalism, or the D-formalism, in short. This formalism exploits the fact that the renormalized Green functions may be calculated by displacing by an infinite amount the renormalized fields and parameters of the theory with respect to the unrenormalized ones. With the help of this formalism, we are able to obtain the precise form of the deformations induced to the Nielsen identities after renormalization, and thus derive the exact dependence of the renormalized Green functions on the renormalized gauge-fixing parameter to all orders. As a particular non-trivial example, we calculate the gauge-dependence of tanβ at two loops in the framework of an Abelian Higgs model, using a gauge-fixing scheme that preserves the Higgs-boson low-energy theorem for off-shell Green functions. Various possible applications and future directions are briefly discussed.