We investigate the interaction of strongly correlated electrons with phonons in the frame of the Hubbard-Holstein model. The electron-phonon interaction is considered to be strong and is an important parameter of the model besides the Coulomb repulsion of electrons and band filling. This interaction with the nondispersive optical phonons has been transformed to the problem of mobile polarons by using the canonical transformation of Lang and Firsov We discuss in particular the case for which the on-site Coulomb repulsion is exactly cancelled by the phonon-mediated attractive interaction and suggest that polarons exchanging phonon clouds can lead to polaron pairing and superconductivity. It is then the frequency of the collective mode of phonon clouds being larger than the bare frequency, which determines the superconducting transition temperature.The aim of the present paper is to gain further insight into the mutual influence of strong onsite Coulomb repulsion and strong electron-phonon interaction by using the single-band HubbardHolstein model and a recently developed diagrammatic approach 1-4. For simplicity we consider coupling to dispersionless phonons only, although this might not be the most interesting case with respect to superconductivity. However, previous investigations 5-7 have shown that the HubbardHolstein model 8-9 constitutes a formidable problem of its own. Other authors have also intensively studied this model Hamiltonian 10-13.Because the interactions between electrons and electrons and phonons are strong, we include the Coulomb repulsion in the zero-order Hamiltonian and apply the canonical transformation of Lang and Firsov 14 in order to eliminate the linear electron-phonon interaction. In the strong electronphonon coupling limit the resulting Hamiltonian of hopping polarons (i.e., hopping electrons surrounded by clouds of phonons) can lead to an attractive interaction among electrons being mediated by the phonons. In this limit the chemical potential, on-site Coulomb energy as well as the frequency of the collective mode of phonon clouds (which is much larger than the bare frequency of the Einstein oscillators) are strongly renormalized 7,15,16 affecting the dynamical properties of the polarons and the character of the superconducting transition. This will be discussed by assuming that renormalized on-site Coulomb repulsion and attractive electron-electron interaction completely cancel each other. We suggest that the resulting superconducting state with polaronic Cooper pairs is mediated by the exchange of phonon clouds during the hopping processes of the electrons.The initial Hamiltonian of correlated electrons coupled to optical phonons with bare frequency 0

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We propose a diagram theory around the atomic limit for the single-impurity Anderson model in which the strongly correlated impurity electrons hybridize with free (uncorrelated) conduction electrons. Using this diagram approach, we prove a linked-cluster theorem for the vacuum diagrams and derive Dyson-type equations for localized and conduction electrons and the corresponding equations for mixed propagators. The system of equations can be closed by summing an infinite series of ladder diagrams containing irreducible Green's functions. The result allows discussing resonances associated with quantum transitions at the impurity site.