A model for the LO-phonon-related structure observed in the luminescence above the gap of InP is presented. The corresponding exciton-phonon quasiparticle spectrum is calculated for zinc-blende-type semiconductors using a Green's-function formalism. It is shown that resonances may appear due to the interaction of the exciton continuum with excitations involving a 1s-exciton state plus a LO phonon. The corresponding electron-hole nonequilibrium distribution function is derived by solving the master equation, which depends on the rate of scattering by acoustic and optical phonons. These results enable the evaluation of the dependence of the luminescence intensity on light frequency and temperature. Explicit calculations are presented for InP, CdTe, and GaAs. In the case of InP they reproduce rather well the experimental luminescence profile observed above the gap and its dependence on temperature. The calculations explain why a similar structure has not been observed in the luminescence spectra of GaAs and CdTe.48851875196