  We adapt the generalized Hartree-Fock-Bogoliubov (HFB) method to an
interacting many-phonon system free of impurities. The many-phonon system is
obtained from applying the Lee-Low-Pine (LLP) transformation to the Frohlich
model which describes a mobile impurity coupled to noninteracting phonons. We
specialize our general HFB description of the Frohlich polaron to Bose polarons
in quasi-1D cold atom mixtures. The LLP transformed many-phonon system
distinguishes itself with an artificial phonon-phonon interaction which is very
different from the usual two-body interaction. We use the quasi-one-dimensional
model, which is free of an ultraviolet divergence that exists in higher
dimensions, to better understand how this unique interaction affects polaron
states and how the density and pair correlations inherent to the HFB method
conspire to create a polaron ground state with an energy in good agreement with
and far closer to the prediction from Feynman's variational path integral
approach than mean-field theory where HFB correlations are absent.
