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Fuchsian holonomic sequences

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Many sequences that arise in combinatorics and the analysis of algorithms turn out to be holonomic (note that some authors prefer the terminology D-finite). In this paper, we study various basic algorithmic problems for such sequences \((f_n)_{n \in {\mathbb {N}}}\): how to compute their asymptotics for large n? How to evaluate \(f_n\) efficiently for large n and/or large precisions p? How to decide whether \(f_n > 0\) for all n? We restrict our study to the case when the generating function \(f = \sum _{n \in {\mathbb {N}}} f_n z^n\) satisfies a Fuchsian differential equation (often it suffices that the dominant singularities of f be Fuchsian). Even in this special case, some of the above questions are related to long-standing problems in number theory. We will present algorithms that work in many cases and we carefully analyze what kind of oracles or conjectures are needed to tackle the more difficult cases.

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Notes

  1. If f has no singularities in \({\mathbb {C}}\), then the assumption that f is Fuchsian at infinity implies that f is actually a polynomial. In what follows, we will discard this trivial case and assume that f has at least one singularity in \({\mathbb {C}}\).

  2. The classical Mellin transform uses a straight integration path from \(z = 0\) to \(+ \infty \) instead of a Hankel contours around \(\alpha _k\). We will say that our Mellin integral is “based at \(\alpha _k\)”. The use of a Hankel contours extends the definition to the case when f is not integrable at \(\alpha _k\). In the context of difference equations, certain authors prefer the terminology “Laplace transform”, “Pincherle transform”, or “Nörlund transform” instead of “Mellin transform”.

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Acknowledgements

We are grateful to Marc Mezzarobba, Ruiwen Dong, and an anonymous referee for comments, suggestions, references, and helpful discussions.

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  1. CNRS, École polytechnique, Institut Polytechnique de Paris, Laboratoire d’informatique de l’École polytechnique (LIX, UMR 7161), 1, rue Honoré d’Estienne d’Orves, Bâtiment Alan Turing, CS35003, Palaiseau, 91120, France

    Joris van der Hoeven

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Correspondence to Joris van der Hoeven.

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van der Hoeven, J. Fuchsian holonomic sequences. AAECC (2023). https://doi.org/10.1007/s00200-023-00616-4

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