  This article is devoted to deduce the expression of the Green's function
related to a general constant coefficients fractional difference equation
coupled to Dirichlet conditions. In this case, due to the points where some of
the fractional operators is applied, we are in presence of a implicit
fractional difference equation. Such property makes it more complicated to
calculate and manage the expression of the Green's function. Such expression,
on the contrary to the explicit case where it follows from finite sums, is
deduced from series of infinity terms. Such expression will be deduced from the
Laplace transform on the time scales of the integers. Finally, we prove two
existence results for nonlinear problems, via suitable fixed point theorems.
