  A formal expansion for the Green's functions of an interacting quantum field
theory in a parameter that somehow encodes its "distance" from the
corresponding non-interacting one was introduced more than thirty years ago,
and has been recently reconsidered in connection with its possible application
to the renormalization of non-hermitian theories. Besides this new and
interesting application, this expansion has special properties already when
applied to ordinary (i.e. hermitian) theories, and in order to disentangle the
peculiarities of the expansion itself from those of non-hermitian theories, it
is worth to push further the investigation limiting first the analysis to
ordinary theories. In the present work we study some aspects related to the
renormalization of a scalar theory within the framework of such an expansion.
Due to its peculiar properties, it turns out that at any finite order in the
expansion parameter the theory looks as non-interacting. We show that when
diagrams of appropriate classes are resummed, this apparent drawback disappears
and the theory recovers its interacting character. In particular we have seen
that with a certain class of diagrams, the weak-coupling expansion results are
recovered, thus establishing a bridge between the two expansions.
