  We investigate the momentum distribution function of a single distinguishable
impurity particle which formed a polaron state in a gas of either free fermions
or Tonks-Girardeau bosons in one spatial dimension. We obtain a Fredholm
determinant representation of the distribution function for the Bethe ansatz
solvable model of an impurity-gas $\delta$-function interaction potential at
zero temperature, in both repulsive and attractive regimes. We deduce from this
representation the fourth power decay at a large momentum, and a weakly
divergent (quasi-condensate) peak at a finite momentum. We also demonstrate
that the momentum distribution function in the limiting case of infinitely
strong interaction can be expressed through a correlation function of the
one-dimensional impenetrable anyons.
