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

On Rational and Hypergeometric Solutions of Linear Ordinary Difference Equations in $\Pi\mathbf\Sigma^*$-field extensions