  We develop a systematic perturbation theory for the quasiparticle properties
of a single impurity immersed in a Bose-Einstein condensate. Analytical results
are derived for the impurity energy, the effective mass, and residue to third
order in the impurity-boson scattering length. The energy is shown to depend
logarithmically on the scattering length to third order, whereas the residue
and effective mass are given by analytical power series. When the boson-boson
scattering length equals the boson-impurity scattering length, the energy has
the same structure as that of a weakly interacting Bose gas, including terms of
the Lee-Huang-Yang and fourth order logarithmic form. Our results, which cannot
be obtained within the canonical Fr{\"o}hlich model of an impurity interacting
with phonons, provide valuable benchmarks for many-body theories and for
experiments.
