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On Positivity and Minimality for Second-Order Holonomic Sequences

Authors George Kenison, Oleksiy Klurman, Engel Lefaucheux, Florian Luca, Pieter Moree , Joël Ouaknine , Markus A. Whiteland , James Worrell

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Author Details

George Kenison
  • Institute for Logic and Computation, The Technical University of Vienna, Austria
Oleksiy Klurman
  • School of Mathematics, University of Bristol, UK
  • Max Planck Institute for Mathematics, Bonn, Germany
Engel Lefaucheux
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Florian Luca
  • School of Mathematics, University of the, Witwatersrand, Johannesburg, South Africa
  • Research Group in Algebraic Structures & , Applications, King Abdulaziz University, Riyadh, Saudi Arabia
  • Centro de Ciencias Matemáticas UNAM, Morelia, Mexico
Pieter Moree
  • Max Planck Institute for Mathematics, Bonn, Germany
Joël Ouaknine
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Markus A. Whiteland
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
James Worrell
  • Department of Computer Science, University of Oxford, UK


This work was partly carried out during a visit of Florian Luca at the Max Planck Institute for Software Systems in Saarbrücken, Germany from September 2020 to March 2021. He thanks the institution for its hospitality and excellent working conditions. The authors would like to thank the anonymous referees for their detailed comments, which have led to significant improvements and clarifications in the final version of this paper.

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George Kenison, Oleksiy Klurman, Engel Lefaucheux, Florian Luca, Pieter Moree, Joël Ouaknine, Markus A. Whiteland, and James Worrell. On Positivity and Minimality for Second-Order Holonomic Sequences. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 67:1-67:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


An infinite sequence ⟨u_n⟩_n of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each u_n ≥ 0, and minimal if, given any other linearly independent sequence ⟨v_n⟩_n satisfying the same recurrence relation, the ratio u_n/v_n → 0 as n → ∞. In this paper we give a Turing reduction of the problem of deciding positivity of second-order holonomic sequences to that of deciding minimality of such sequences. More specifically, we give a procedure for determining positivity of second-order holonomic sequences that terminates in all but an exceptional number of cases, and we show that in these exceptional cases positivity can be determined using an oracle for deciding minimality.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Holonomic sequences
  • Minimal solutions
  • Positivity Problem


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