This paper reports the observation of phonon sidebands in the photoluminescence spectra of Be acceptor-doped GaAs/AlAs multiple quantum wells. The intensity and energetic positions of the sideband lines are investigated experimentally for several quantum wells having various doping concentrations and photoluminescence excitation intensities. Theoretical analysis of a sideband-related lineshape, considering their energy position and impurity-induced spectra, has shown that phonon satellites can be attributed to free-electron -Be acceptor transitions involving longitudinal optical phonons of the GaAs-the host material of the studied quantum wells. The Huang-Rhys factor which determines the distribution of luminescence intensities between the phonon replicas and the main no-phonon peak was examined both experimentally and theoretically by varying the quantum well width. Thus, it has been found that this factor increases monotonically from 0.052 to 0.11 as the width of the quantum well decreases from 20 nm to 5 nm. The dependence of the Huang-Rhys factor on the width of the quantum well for a free-to-acceptor recombination was calculated applying the fractional-dimensional space approach. The proposed model adequately describes the experimentally determined dependence of the Huang-Rhys factor