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arXiv · 2005.04944

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

Sergei A. Abramov·Manuel Bronstein·Marko Petkovšek·Carsten Schneider
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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 $\Pi\Sigma^*$-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.

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Sergei A. Abramov, Manuel Bronstein, Marko Petkovšek, Carsten Schneider. 2020-05-11. On Rational and Hypergeometric Solutions of Linear Ordinary Difference Equations in $\Pi\mathbf\Sigma^*$-field extensions. https://arxiv.org/abs/2005.04944

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