Abstract The Hubbard-Holstein model after canonical Lang-Firsov transformation is investigated. Two peculiarities of the model are taken into account: a) the strong electron correlations and b) the fact that the hoppings of polarons between lattice sites are accompanied by an indefinite number of phonons. A new diagram technique based on the classification of irreducible many-particle Green functions or Kubo cumulants is developed. Phonon renormalization of electronic functions is considered. A theorem about connected vacuum diagrams is established, and Dyson equation for electron Green function is obtained.