  We present a complete algorithm that computes all hypergeometric solutions of
homogeneous linear difference equations and rational solutions of parameterized
linear difference equations in the setting of $\Pi\Sigma^*$-fields. More
generally, we provide a flexible framework for a big class of difference fields
that is built by a tower of $\Pi\Sigma^*$-field extensions over a difference
field that satisfies certain algorithmic properties. As a consequence one can
compute all solutions in terms of indefinite nested sums and products that
arise within the components of a parameterized linear difference equation, and
one can find all hypergeometric solutions that are defined over the arising
sums and products of a homogeneous linear difference equation.
