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Fast Algorithms for Refined Parameterized Telescoping in Difference Fields

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Computer Algebra and Polynomials

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8942))

Abstract

Parameterized telescoping (including telescoping and creative telescoping) and refined versions of it play a central role in the research area of symbolic summation. In 1981 Karr introduced \(\varPi \varSigma \)-fields, a general class of difference fields, that enables one to consider this problem for indefinite nested sums and products covering as special cases, e.g., the (\(q\)–)hypergeometric case and their mixed versions. This survey article presents the available algorithms in the framework of \(\varPi \varSigma \)-extensions and elaborates new results concerning efficiency.

C. Schneider—Supported by the Austrian Science Fund (FWF) grants P20347-N18 and SFB F50 (F5009-N15) and by the EU Network LHCPhenoNet PITN-GA-2010-264564.

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Notes

  1. 1.

    Karr’s \(\varSigma \)-extensions [27] are given by generators with \(\sigma (t)=\alpha \,t+\beta \) with extra conditions on \(\alpha \). For simplicity, we work with \(\varSigma ^*\)-extensions that are relevant in symbolic summation.

  2. 2.

    \(({\mathbb {K}},{\sigma })\) is \(\sigma \)-computable (see Definition 5) iff \(\mathbb {K}\) is \(\sigma \)-computable (see Definition 4); we refer to [27, 30].

  3. 3.

    To execute the steps above one needs the property that \(({\mathbb {G}},{\sigma })\) is FPLDE-solvable. However, in the following we are only interested in the the exploration of the possible reduction processes with the corresponding reduction vectors without the need to calculate them explicitly. We therefore drop the FPLDE-solvability in the statements below.

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Acknowledgments

I would like to thank the referee for the very careful reading and the valuable suggestions.

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  1. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Carsten Schneider

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  1. University of Cantabria, Santander, Spain

    Jaime Gutierrez

  2. Ricam Linz, Linz, Austria

    Josef Schicho

  3. University of Caen, Caen, France

    Martin Weimann

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Schneider, C. (2015). Fast Algorithms for Refined Parameterized Telescoping in Difference Fields. In: Gutierrez, J., Schicho, J., Weimann, M. (eds) Computer Algebra and Polynomials. Lecture Notes in Computer Science(), vol 8942. Springer, Cham. https://doi.org/10.1007/978-3-319-15081-9_10

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