The diagrammatic expansion in the real-time domain of the one-electron and electron-hole propagators for the small-polaron problem is presented. For this purpose, special graphs consisting of fermion lines, multiphonon interaction lines, and migration vertices are used. The spectral function of the small-polaron propagator is calculated for two cases. First, the interaction with a narrow band of optical modes, and second, the interaction with longitudinal acoustic modes, is considered. The expressions for the small-polaron mobility in the hopping region are also derived for both models.