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Hermite Reduction for D-finite Functions via Integral Bases

Published: 24 July 2023 Publication History

Abstract

Trager’s Hermite reduction solves the integration problem for algebraic functions via integral bases. A generalization of this algorithm to D-finite functions has so far been limited to the Fuchsian case. In the present paper, we remove this restriction and propose a reduction algorithm based on integral bases that is applicable to arbitrary D-finite functions.

References

[1]
[1] https://github.com/LixinDu/HermiteReduction.
[2]
[2] Sergei A Abramov and Mark Van Hoeij. Integration of solutions of linear functional equations. Integral Transforms and Special Functions, 8(1-2):3–12, 1999.
[3]
[3] Shayea Aldossari. Algorithms for Simplifying Differential Equations. 2020. PhD thesis.
[4]
[4] Gert Almkvist and Doron Zeilberger. The method of differentiating under the integral sign. Journal of Symbolic Computation, 10:571–591, 1990.
[5]
[5] Alin Bostan, Shaoshi Chen, Frédéric Chyzak, and Ziming Li. Complexity of creative telescoping for bivariate rational functions. In Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, page 203–210, New York, NY, USA, 2010. ACM.
[6]
[6] Alin Bostan, Shaoshi Chen, Frédéric Chyzak, Ziming Li, and Guoce Xin. Hermite reduction and creative telescoping for hyperexponential functions. In Proceedings of the 2013 International Symposium on Symbolic and Algebraic Computation, pages 77–84, New York, NY, USA, 2013. ACM.
[7]
[7] Alin Bostan, Frédéric Chyzak, Pierre Lairez, and Bruno Salvy. Generalized Hermite reduction, creative telescoping and definite integration of D-finite functions. In Proceedings of the 2018 International Symposium on Symbolic and Algebraic Computation, pages 95–102, New York, NY, USA, 2018. ACM.
[8]
[8] Shaoshi Chen, Lixin Du, and Manuel Kauers. Lazy Hermite reduction and creative telescoping for algebraic functions. In Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation, page 75–82, New York, NY, USA, 2021. ACM.
[9]
[9] Shaoshi Chen, Hui Huang, Manuel Kauers, and Ziming Li. A modified Abramov-Petkovsek reduction and creative telescoping for hypergeometric terms. In Proceedings of the 2015 International Symposium on Symbolic and Algebraic Computation, pages 117–124, New York, NY, USA, 2015. ACM.
[10]
[10] Shaoshi Chen, Manuel Kauers, and C. Koutschan. Reduction-based creative telescoping for algebraic functions. In Proceedings of the 2016 International Symposium on Symbolic and Algebraic Computation, pages 175–182, New York, NY, USA, 2016. ACM.
[11]
[11] Shaoshi Chen, Mark van Hoeij, Manuel Kauers, and Christoph Koutschan. Reduction-based creative telescoping for fuchsian D-finite functions. Journal of Symbolic Computation, 85:108–127, 2018.
[12]
[12] Frédéric Chyzak. An extension of Zeilberger’s fast algorithm to general holonomic functions. Discrete Mathematics, 217:115–134, 2000.
[13]
[13] Lixin Du. Generalized Integral Bases and Applications in Creative Telescoping. 2022. PhD thesis.
[14]
[14] Erdal Imamoglu. Algorithms for Solving Linear Differential Equations with Rational Function Coefficients. 2017. PhD thesis.
[15]
[15] Erdal Imamoglu and Mark van Hoeij. Computing hypergeometric solutions of second order linear differential equations using quotients of formal solutions and integral bases. Journal of Symbolic Computation, 83:254–271, 2017.
[16]
[16] Edward L. Ince. Ordinary Differential Equations. Dover, 1926.
[17]
[17] Manuel Kauers and Christoph Koutschan. Integral D-finite functions. In Proceedings of the 2015 International Symposium on Symbolic and Algebraic Computation, pages 251–258, New York, NY, USA, 2015. ACM.
[18]
[18] Manuel Kauers and Peter Paule. The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates. Springer Publishing Company, Incorporated, 1st edition, 2011.
[19]
[19] Mikhail Vasil’evich Ostrogradsky. De l’intégration des fractions rationnelles. Bull. de la classe physico-mathématique de l’Acad. Impériale des Sciences de Saint-Pétersbourg, 4:145–167, 286–300, 1845.
[20]
[20] Barry M. Trager. Integration of Algebraic Functions. 1984. PhD thesis.
[21]
[21] Joris van der Hoeven. Constructing reductions for creative telescoping: the general differentially finite case. Applicable Algebra in Engineering, Communication and Computing, 32(5):575–602, nov 2021.
[22]
[22] Doron Zeilberger. A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics, 32:321–368, 1990.

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    ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation
    July 2023
    567 pages
    ISBN:9798400700392
    DOI:10.1145/3597066
    This work is licensed under a Creative Commons Attribution International 4.0 License.

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    Association for Computing Machinery

    New York, NY, United States

    Publication History

    Published: 24 July 2023

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    Author Tags

    1. Additive decomposition
    2. creative telescoping
    3. symbolic integration

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    • Research-article
    • Research
    • Refereed limited

    Funding Sources

    • Austrian FWF grants
    • CAS Project for Young Scientists in Basic Research
    • NSFC grants
    • National Key Research and Development Project
    • Fund of the Youth Innovation Promotion Association

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    ISSAC 2023

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    Overall Acceptance Rate 395 of 838 submissions, 47%

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