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The Ultimate Signs of Second-Order Holonomic Sequences

Authors Fugen Hagihara , Akitoshi Kawamura

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ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
Keywords
  • Holonomic sequences
  • ultimate signs
  • Skolem Problem
  • Positivity Problem

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Abstract

A real-valued sequence f = {f(n)}_{n ∈ ℕ} is said to be second-order holonomic if it satisfies a linear recurrence f (n + 2) = P (n) f (n + 1) + Q (n) f (n) for all sufficiently large n, where P, Q ∈ ℝ(x) are rational functions. We study the ultimate sign of such a sequence, i.e., the repeated pattern that the signs of f (n) follow for sufficiently large n. For each P, Q we determine all ultimate signs that f can have, and show how they partition the space of initial values of f. This completes the prior work by Neumann, Ouaknine and Worrell, who have settled some restricted cases. As a corollary, it follows that when P, Q have rational coefficients, f either has an ultimate sign of length 1, 2, 3, 4, 6, 8 or 12, or never falls into a repeated sign pattern. We also give a partial algorithm that finds the ultimate sign of f (or tells that there is none) in almost all cases.

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Fugen Hagihara and Akitoshi Kawamura. The Ultimate Signs of Second-Order Holonomic Sequences. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 159:1-159:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.159

Author Details

Fugen Hagihara
  • Graduate School of Science, Kyoto University, Japan
Akitoshi Kawamura
  • Research Institute for Mathematical Sciences, Kyoto University, Japan

Funding

This work was supported by JSPS KAKENHI Grant Numbers JP18H03203, JP23K28036, JP25KJ1559 and by JST SPRING Grant Number JPMJSP2110.

Acknowledgements

We thank Kohki Baku at Faculty of Science, Kyoto University, for helping us find the explicit formula (8) in Example 5. We also thank the anonymous referees for knowledgeable comments, which helped us clarify the explanation on previous works and our results.

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