  We use a lattice Green function approach to study the stationary modes of a
linear/nonlinear (Kerr) impurity embedded in a periodic one-dimensional lattice
where we replace the standard discrete Laplacian by a fractional one. The
energies and the mode profiles are computed in closed form, for different
fractional exponents and different impurity strengths. The energies of the
impurity mode lie outside the linear band whose bandwidth decreases steadily as
the fractional exponent decreases. For any fractional exponent values, there is
always a single bound state for the linear impurity while for the nonlinear
(Kerr) case, up to two bound states are possible, for impurity strengths above
certain threshold. The energy of the linear mode (or that of the upper energy
nonlinear one), becomes directly proportional to the impurity strength at large
impurity strengths. The transmission of plane waves is also computed in closed
form for several fractional exponents, and various impurity strengths. We
observe that fractionality tends to increase the overall transmission. The
selftrapping transition for the nonlinear impurity shifts to lower nonlinearity
values as the fractional exponent is decreased. In both cases, linear and
nonlinear, we observe a form of trapping at zero impurity strength, which can
be explained by the near-degeneracy of the spectrum in the limit of a small
fractional exponent.
