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The frequency and wave-vector-dependent memory function in the longitudinal conductivity tensor of weakly interacting electronic systems is calculated by using the approach based on the quantum transport equations. It is shown, for the first time consistently, that there is a close relation between the single-electron self-energy, the electron-hole pair self-energy, and the memory function. It is also shown in which way singular long-range Coulomb interactions, together with other q approx 0 scattering processes, drop out of both the memory function and the related transport equations. The theory is illustrated on heavily doped graphene, which is the prototype of weakly interacting single-band electron-phonon systems. A steplike increase of the width of the quasiparticle peak in angle-resolved photoemission spectra at frequencies of the order of the frequency of in-plain optical phonons is shown to be consistent with similar behavior of the intraband plasmon peak in energy loss spectroscopy spectra. Both anomalies can be understood as a direct consequence of weak electron scattering from in-plane optical phonons.