  We present a formal derivation of the many-body perturbation theory for a
system of electrons and bosons subject to a nonlinear electron-boson coupling.
The interaction is treated at an arbitrary high order of bosons scattered. The
considered Hamiltonian includes the well-known linear coupling as a special
limit. This is the case, for example, of the Holstein and Fr\"{o}hlich
Hamiltonians. Indeed, whereas linear coupling have been extensively studied,
the scattering processes of electrons with multiple bosonic quasiparticles are
largely unexplored. We focus here on a self-consistent theory in terms of
dressed propagators and generalize the Hedin's equations using the Schwinger
technique of functional derivatives. The method leads to an exact derivation of
the electronic and bosonic self-energies, expressed in terms of a new family of
vertex functions, high order correlators and bosonic and electronic mean-field
potentials. In the electronic case we prove that the mean-field potential is
the $n$th-order extension of the well-known Debye-Waller potential. We also
introduce a bosonic mean-field potential entirely dictated by nonlinear
electron-boson effects. The present scheme, treating electrons and bosons on an
equal footing, demonstrates the full symmetry of the problem. The vertex
functions are shown to have purely electronic and bosonic character as well as
a mixed electron-boson one. These four vertex functions are shown to satisfy a
generalized Bethe-Salpeter equation. Multi bosons response functions are also
studied and explicit expressions for the two and the three bosons case are
given.
