Skip to main content
SpringerLink
Log in

Fuchsian holonomic sequences

  • Original Paper
  • Published:
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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

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”.

References

  1. van der Hoeven, J.: The Jolly Writer. Your Guide to GNU TeXmacs, Scypress (2020)

  2. Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comp. Appl. Math. 32, 321–368 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Flajolet, P., Sedgewick, R.: Analytic combinatorics. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  4. Kenison, G., Lipton, R., Ouaknine, J., Worrell, J.: On the Skolem problem and prime powers. In Proc. of the 45th Intern. Symp. on Symbolic and AlgebraicComputation, ISSAC ’20, pp. 289–296. ACM, (2020)

  5. Nosan, K., Pouly, A., Shirmohammadi, M., Worrell, J.: The membership problem for hypergeometric sequences with rational parameters. In Proc. ISSAC ’22, pp. 381–389. (2022)

  6. Melczer, S., Mezzarobba, M.: Sequence positivity through numeric analytic continuation: uniqueness of the Canham model for biomembranes. Combinatorial Theory (2022). https://doi.org/10.5070/C62257847

    Article  MathSciNet  MATH  Google Scholar 

  7. Chudnovsky, D. V., Chudnovsky, G. V.: Computer algebra in the service of mathematical physics and number theory (Computers in mathematics, Stanford, CA, 1986). In Lect. Notes in Pure and Applied Math., vol. 125, pp. 109–232. New-York, (1990)

  8. van der Hoeven, J.: Fast evaluation of holonomic functions. TCS 210, 199–215 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. van der Hoeven, J.: Fast evaluation of holonomic functions near and in singularities. JSC 31, 717–743 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Mezzarobba, M.: Autour de l’évaluation numérique des fonctions D-finies. PhD thesis, École polytechnique, Palaiseau, France

  11. Mezzarobba, M.: Rigorous multiple-precision evaluation of D-Finite functions in SageMath. Technical Report, HAL, (2016). https://hal.archives-ouvertes.fr/hal-01342769

  12. Dong, R.: Computing error bounds for asymptotic expansions of regular P-recursive sequences. Master’s thesis, École polytechnique,: Unpublished. Supervised by S. Melczer and M, Mezzarobba (2021)

  13. Dong, R., Melczer, S., Mezzarobba, M.: Computing error bounds for asymptotic expansions of regular P-recursive sequences. Technical Report, ArXiv, 2022. arXiv:https://arxiv.org/abs/2212.11742

  14. Gerhold, S., Kauers, M.: A procedure for proving special function inequalities involving a discrete parameter. In Proc. of the 2005 Intern.Symp. on Symbolicand AlgebraicComputation, ISSAC 2005, pp. 156–162. ACM, (2005)

  15. Kauers, M., Pillwein, V.: When can we detect that a p-finite sequence is positive? In Proc. of the 2010 Intern. Symp. on Symbolic and Algebraic Computation, ISSAC 2010, pp. 195–201. ACM, (2010)

  16. Pincherle, S.: Sur la génération de systèmes récurrents au moyen d’une équation linéaire différentielle. Acta Math. 16, 341–363 (1892)

    Article  MathSciNet  MATH  Google Scholar 

  17. van der Hoeven, J.: On the computation of limsups. J. Pure Appl. Algebra 117(118), 381–394 (1996)

    MathSciNet  MATH  Google Scholar 

  18. Nörlund, N.E.: Fractions continues et différences réciproques. Acta Math. 34, 1–108 (1910)

    Article  MATH  Google Scholar 

  19. Galbrun, H.: Sur la représentation des solutions d’une équation linéaire aux différences finies pour les grandes valeurs de la variable. Acta Math. 36, 1–68 (1913)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nörlund, N.E.: Vorlesungen über Differenzenrechnung. Springer Verlag, Berlin (1924)

    Book  MATH  Google Scholar 

  21. Galbrun, H.: Sur certaines solutions exceptionnelles d’une équation linéaire aux différences finies. Bull. S.M.F. 49, 206–241 (1921)

    MathSciNet  MATH  Google Scholar 

  22. Duval, A.: Équations aux différences dans le champ complexe. PhD thesis, IRMA, Strasbourg, France, (1984)

  23. Barkatou, M.A.: Contributions à l’étude d’équations différentielleset aux différencesdans le champ complexe. PhD thesis, Institut National Polytechnique de Grenoble, (1989)

  24. Immink, G.K.: On the relation between global properties of linear difference and differential equations with polynomial coefficients. I. J. Diff. Eq. 113, 201–233 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Immink, G.K.: On the relation between global properties of linear difference and differential equations with polynomial coefficients. II. J. Diff. Eq. 128, 168–205 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Immink, G.K.: On the relation between global properties of linear difference and differential equations with polynomial coefficients. Math. Nachr. 200, 59–76 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Fuchs, L.: Zur Theorie der Linearen Differentialgleichungen mit veränderlichen Coefficienten. J. für die reine une angewandte Math. 66(2), 121–160 (1866)

    MATH  Google Scholar 

  28. van der Put, M., Singer, M.: Galois Theory of Linear Differential Equations, volume 328 of Grundlehren der mathematischen Wissenschaften. Springer, (2003)

  29. van der Hoeven, J.: Efficient accelero-summation of holonomic functions. Technical Report, HAL, (2021). https://hal.archives-ouvertes.fr/hal-03154241, corrected version of [25]

  30. Fischler, S., Rivoal, T.: On the values of G-functions. Comment. Math. Helv. 89(2), 313–341 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  32. Th. Skolem. Einige Sätze über gewisse Reihenentwicklungen und exponentiale Beziehungen mit anwendung auf diophantische Gleichungen. Oslo Vid. akad. Skrifter, 1(6), (1933)

  33. Mahler, K.: Eine arithmetische Eigenschaft der Taylor-koeffizienten rationaler Funktionen. Proc. Akad. Wetensch. Amsterdam 38, 50–60 (1935)

    MATH  Google Scholar 

  34. Lech, C.: A note on recurring series. Ark. Mat. 2, 417–421 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  35. Berstel, J., Mignotte, M.: Deux propriétés décidables des suites récurrentes linéaires. Bull. de la S.M.F. 104, 175–184 (1976)

    MATH  Google Scholar 

  36. Ouaknine, J., Worrell, J.: Decision problems for linear recurrence sequences. In Reachability Problems: 6th International Workshop, RP 2012, volume 7550 of Lect.Notesin Comp. Science, pages 21–28. Bordeaux, France, September (2012). Springer-Verlag

  37. Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Illinois J. Math. 6(1), 64–94 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  38. Householder, A.S.: Dandelin, Lobacevskii, or Graeffe. Am. Math. Mon. 66(6), 464–466 (1959)

    MathSciNet  MATH  Google Scholar 

  39. Baker, A.: Linear forms in the logarithms of algebraic numbers (III). Mathematika 14(2), 220–228 (1967)

    Article  MATH  Google Scholar 

  40. van der Hoeven, J.: Zero-testing, witness conjectures and differential diophantine approximation. Technical Report 2001-62, Prépublications d’Orsay, (2001)

  41. Harvey, D., van der Hoeven, J.: Integer multiplication in time \(O (n \log n)\). Ann. Math. 193(2), 563–617 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  42. Brent, R.P.: Fast multiple-precision evaluation of elementary functions. J. ACM 23(2), 242–251 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  43. Boshernitzan, M.D.: Uniform distribution and Hardy fields. J. Anal. Math. 62, 225–240 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  44. Bourbaki, N.: Fonctions d’une variable réelle. Éléments de Mathématiques (Chap. 5). Hermann, 2-nd edition, (1961)

  45. Basu, S., Pollack, R., Roy, M.-F.: Algorithms in real algebraic geometry, volume 10 of Algorithms and Computation in Mathematics. Springer-Verlag, 2-nd edition to appear (2006)

  46. Birkhoff, G.D.: Formal theory of irregular linear difference equations. Acta Math. 54, 205–246 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  47. van der Hoeven, J.: Relax, but don’t be too lazy. JSC 34, 479–542 (2002)

    MathSciNet  MATH  Google Scholar 

  48. van der Hoeven, J.: From implicit to recursive equations. AAECC 30(3), 243–262 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  49. van der Hoeven, J.: Efficient accelero-summation of holonomic functions. JSC 42(4), 389–428 (2007)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  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

Authors
  1. Joris van der Hoeven
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joris van der Hoeven.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This document has been written using GNU TEXmacs [1].

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00200-023-00616-4

Search

Navigation