In some semiconducting materials, like transition-metal oxides [1-3] and organic semiconductors [4,5], one may easily imagine impurities characterized by a short-ranged electron-phonon coupling, i.e. polaronic impurities, as a source of scattering for the electrons. Although dynamic electron scattering on a phonon degree of freedom has been investigated before in the context of the electron tunneling across microscopic junctions, nanowires and quantum dots [6,7], to the best of our knowledge, the effect of polaronic impurities on the charge transport in bulk systems has not been studied. Thus, we described, in a close form, electron scattering on a phonon degree of freedom within the Green function formalism, establishing a connection between the T-matrix and Feynman diagrams to all orders in electron-phonon interaction. Within the formalism, we calculated electron partial scattering cross sections for elastic and inelastic processes and analyzed dependence of the electron mobility on the impurity strength, but also its temperature dependence for some model parameters. In addition, structure of bound electron states is analyzed for various model parameters.
 A. J. Bosman, H. J. van Daal, Adv. Phys. 19, (1970) 1.
 M. Reticcioli, U. Diebold, G. Kresse, C. Franchini, Handbook of Materials Modeling: Applications: Current and Emerging Materials (eds. W. Andreoni, S. Yip, Springer, 2019).
 F. De Angelis, C. Di Valentin, S. Fantacci, A. Vittadini, A. Selloni, Chem. Rev. 114 (2014) 9708.
 F. Ortmann, S. Roche, Phys. Rev. B 84 (2011) 180302.
 I. Salzmann, G. Heimel, M. Oehzelt, S. Winkler, N. Koch, Acc. Chem. Res. 49 (2016) 370.
 N. A. Zimbovskaya, M. R. Pederson, Phys. Rep. 509 (2011) 1.
 N. B. Zhitenev, H. Meng, Z. Bao, Phys. Rev. Lett. 88 (2002) 226801.