Refined telescoping algorithms in $R\Pi\Sigma$-extensions to reduce the degrees of the denominators

TL;DR

This paper introduces refined telescoping algorithms within $R ext{-}oldsymbol{ ext{Pi}} ext{-}oldsymbol{ ext{Sigma}}$-extensions to produce simpler sum representations with lower-degree denominators, improving the simplification of nested sums.

Contribution

It develops a general framework for difference ring extensions that reduces the degrees of denominators in nested sums, applicable to $R ext{-}oldsymbol{ ext{Pi}} ext{-}oldsymbol{ ext{Sigma}}$-ring extensions over $oldsymbol{ ext{Pi}} ext{-}oldsymbol{ ext{Sigma}}$-fields.

Findings

Enhanced sum representations with lower-degree denominators.

Applicable to simplifying d'Alembertian and Liouvillian solutions.

Improves the efficiency of telescoping algorithms in difference algebra.

Abstract

We present a general framework in the setting of difference ring extensions that enables one to find improved representations of indefinite nested sums such that the arising denominators within the summands have reduced degrees. The underlying (parameterized) telescoping algorithms can be executed in $R\Pi\Sigma$-ring extensions that are built over general $\Pi\Sigma$-fields. An important application of this toolbox is the simplification of d'Alembertian and Liouvillian solutions coming from recurrence relations where the denominators of the arising sums do not factor nicely.