Intermultiplet transitions involving emission (or absorption) of one photon and n phonons are theoretically investigated in trivalent rare earth (RE) ions in crystals. Transitions are induced by joint action of electromagnetic field and high-order optical anharmonicity (M-process). Hence, this work extends Judd's study of one photon-one phonon transitions to the multiphonon case. The processes under consideration give rise to the Stokes and anti-Stokes multiphonon sidebands of light emission or absorption. Besides, these processes can significantly increase the rate of intermultiplet transitions in the presence of stimulated emission. As far as we know the problem of multiphonon sidebands have not been considered previously in the theory of M-processes. Here we obtained general expression for transition probabilities. In doing this we used the crystal field model that takes into account both Coulomb and non-Coulomb interactions of 4f-electrons with ligands. (Previously we used this model for the calculation of multiphonon rates in RE ions.) The "electronic part" of obtained expression is presented exactly in the same form as the well-known Judd-Ofelt expression for radiative transition probability between multiplets. A simplified form of general expression is proposed. Other mechanisms of photon-phonon transitions are briefly discussed. © 1999 Elsevier Science B.V. All rights reserved.