  We investigate the collective excitations of $^3$He-$^4$He mixture films at
zero temperature within random phase approximation and linear response theory.
In low concentration region of $^3$He, a level repulsion between zero sound and
third sound mode is derived, which opens the possibility to observe quantum
mechanical coherence between $^3$He particle-hole pair and the condensate of
$^4$He. We also investigate the $^3$He-$^4$He vertex corrections in ladder
approximation and show that the third branch, the combined mode of fermionic
particle-hole pair and the third sound quanta, provides a unique correction to
Landau $f$ function. Some implications to the fermion-boson mixture of alkali
atoms in a potential trap is discussed.
