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Solving Linear Recurrence Equations with Polynomial Coefficients

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Computer Algebra in Quantum Field Theory

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

Summation is closely related to solving linear recurrence equations, since an indefinite sum satisfies a first-order linear recurrence with constant coefficients, and a definite proper-hypergeometric sum satisfies a linear recurrence with polynomial coefficients. Conversely, d’Alembertian solutions of linear recurrences can be expressed as nested indefinite sums with hypergeometric summands. We sketch the simplest algorithms for finding polynomial, rational, hypergeometric, d’Alembertian, and Liouvillian solutions of linear recurrences with polynomial coefficients, and refer to the relevant literature for state-of-the-art algorithms for these tasks. We outline an algorithm for finding the minimal annihilator of a given P-recursive sequence, prove the salient closure properties of d’Alembertian sequences, and present an alternative proof of a recent result of Reutenauer’s that Liouvillian sequences are precisely the interlacings of d’Alembertian ones.

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Notes

  1. 1.

    C-recursive sequences are also called linear recurrent (or: recurrence) sequences. This neglects sequences satisfying linear recurrences with non-constant coefficients, and may lead to confusion.

  2. 2.

    A hypergeometric sequence is also called a hypergeometric term, because the nth term of a hypergeometric series, considered as a function of n, is a hypergeometric sequence in our sense.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000, Ljubljana, Slovenia

    Marko Petkovšek

  2. Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva 6, SI-1000, Ljubljana, Slovenia

    Helena Zakrajšek

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  1. Marko Petkovšek
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  2. Helena Zakrajšek
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Correspondence to Marko Petkovšek .

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Editors and Affiliations

  1. RISC, Johannes Kepler University, Linz, Austria

    Carsten Schneider

  2. DESY, Zeuthen, Germany

    Johannes Blümlein

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Petkovšek, M., Zakrajšek, H. (2013). Solving Linear Recurrence Equations with Polynomial Coefficients. In: Schneider, C., Blümlein, J. (eds) Computer Algebra in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1616-6_11

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  • DOI: https://doi.org/10.1007/978-3-7091-1616-6_11

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  • Online ISBN: 978-3-7091-1616-6

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