  We study the dynamical response of ultracold fermion-boson mixture in the
Bogoliubov regime, where the interactions between fermionic impurities and
bosonic excitations (phonons) are described by an effective Frohlich model
under the Bogoliubov approximation. A characteristic suppression of the
single-particle spectral weight is found in the small momentum region where the
impurity band and phonon mode intersect. Using diagrammatic technique we
compute the Bragg spectra as well as the momentum dependent force-force
correlation function. We fnd that both of them are heavily affected by the
spectral weight suppression effect at low impurity densities in both 1D and 2D
systems. We show that the the spectral weight suppression feature in Bragg
spectra, which was previously found in the quantum Monte Carlo simulations and
which cannot be recovered by the random phase approximation, can be accurately
reproduced with the help of vertex corrections.
