LIPIcs.ICALP.2021.99.pdf
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Decision Problems for Second-Order Holonomic Recurrences
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48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Colloquium on Automata, Languages, and Programming (ICALP) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2021-07-02
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Eike Neumann, Joël Ouaknine, and James Worrell. Decision Problems for Second-Order Holonomic Recurrences. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 99:1-99:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.99
Abstract
We study decision problems for sequences which obey a second-order holonomic recurrence of the form f(n + 2) = P(n) f(n + 1) + Q(n) f(n) with rational polynomial coefficients, where P is non-constant, Q is non-zero, and the degree of Q is smaller than or equal to that of P. We show that existence of infinitely many zeroes is decidable. We give partial algorithms for deciding the existence of a zero, positivity of all sequence terms, and positivity of all but finitely many sequence terms. If Q does not have a positive integer zero then our algorithms halt on almost all initial values (f(1), f(2)) for the recurrence. We identify a class of recurrences for which our algorithms halt for all initial values. We further identify a class of recurrences for which our algorithms can be extended to total ones.
Subject Classification
ACM Subject Classification
- Mathematics of computing → Discrete mathematics
Keywords
- holonomic sequences
- Positivity Problem
- Skolem Problem
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References
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