  We generalize the application of small polaron theory to ultracold gases of
Ref. [\onlinecite{jaksch_njp1}] to the case of Bose-Fermi mixtures, where both
components are loaded into an optical lattice. In a suitable range of
parameters, the mixture can be described within a Bogoliubov approach in the
presence of fermionic (dynamic) impurities and an effective description in
terms of polarons applies. In the dilute limit of the slow impurity regime, the
hopping of fermionic particles is exponentially renormalized due to polaron
formation, regardless of the sign of the Bose-Fermi interaction. This should
lead to clear experimental signatures of polaronic effects, once the regime of
interest is reached. The validity of our approach is analyzed in the light of
currently available experiments. We provide results for the hopping
renormalization factor for different values of temperature, density and
Bose-Fermi interaction for three-dimensional $^{87}\rm{Rb}-^{40}\rm{K}$
mixtures in optical lattice.
