  The infinite-$U$ Anderson-Holstein impurity model is studied with a focus on
the interplay between the strong electron correlation and the weak
electron-phonon interaction. The slave boson method has been employed in
combination with the large degeneracy expansion (1/N) technique. The charge and
spin susceptibilities and the phonon propagator are obtained in the
approximation scheme where the saddle point configuration and the Gaussian 1/N
fluctuations are taken into account. The spin susceptibility is found not to be
renormalized by electron-phonon interaction, while the charge susceptibility is
renormalized.
  From the renormalized charge susceptibility the Kondo temperature is found to
increase by the electron-phonon interaction. It turns out that the bosonic 1/N
Gaussian fluctuations play a very crucial role, in particular, for the phonon
propagator.
