On rational and hypergeometric solutions of linear ordinary difference equations in ΠΣ⁎-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 ΠΣ⁎-fields. More generally, we provide a flexible framework for a big class of difference fields that are built by a tower of ΠΣ⁎-field extensions over a difference field that enjoys 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 of a homogeneous linear difference equation that are defined over the arising sums and products.