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

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arxiv 2302.03563 v1 pith:MQ3IME5U submitted 2023-02-07 cs.SC

classification cs.SC
keywords denominatorsextensionssigmaalgorithmsarisingdegreesgeneralring
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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.

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  1. Telescoping Algorithms for $\Sigma^*$-Extensions via Complete Reductions

    cs.SC 2025-06 conditional novelty 7.0 of 10

    Complete reductions can be built recursively in towers of Sigma*-extensions, yielding faster refined and parameterized telescoping algorithms for nested harmonic sums.

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