  We study a mixture of one-dimensional bosons and spinless fermions at
incommensurate filling using phenomenological bosonization and Green's
functions techniques. We derive the relation between the parameters of the
microscopic Hamiltonian and macroscopic observables. Galilean invariance
results in extra constraints for the current current interactions. We obtain
the exact exponents for the various response functions, and show that
superfluid fluctuations are enhanced by the effective boson-fermion
density-density interaction and suppressed by the effective boson-fermion
current-current interaction. In the case of a bosonized model with purely
density-density interaction, when the effective boson-fermion density-density
interaction is weak enough, the superfluid exponent of the bosons has a
non-monotonous variation with the ratio of the fermion velocity to the boson
velocity. By contrast, density-wave exponent and the exponent for fermionic
superfluidity are monotonous functions of the velocity ratio.
