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

Abstract

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 $\Pi {\Sigma }^{\ast }$-fields. More generally, we provide a flexible framework for a big class of difference fields that is built by a tower of $\Pi {\Sigma }^{\ast }$-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.

arXiv commentVarious typos have been removed and the presentation has been improved

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