  We investigate confinement-induced resonances in a system composed by a
tightly trapped ion and a moving atom in a waveguide. We determine the
conditions for the appearance of such resonances in a broad region -- from the
"long-wavelength" limit to the opposite case when the typical length scale of
the atom-ion polarisation potential essentially exceeds the transverse
waveguide width. We find considerable dependence of the resonance position on
the atomic mass which, however, disappears in the "long-wavelength and
zero-energy" limit, where the known result for the confined atom-atom
scattering is reproduced. We also derive an analytic and a semi-analytic
formula for the resonance position in the "long-wavelength and zero-energy"
limit and we investigate numerically how the position of the resonance is
affected by a finite atomic colliding energy. Our results, which can be
investigated experimentally in the near future, could be used to determine the
atom-ion scattering length, the temperature of the atomic ensemble in the
presence of an ion impurity, and to control the atom-phonon coupling in a
linear ion crystal in interaction with a quasi one dimensional atomic quantum
gas.
