Decision Problems for Second-Order Holonomic Recurrences
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.
holonomic sequences
Positivity Problem
Skolem Problem
Mathematics of computing~Discrete mathematics
99:1-99:20
Track A: Algorithms, Complexity and Games
Eike
Neumann
Eike Neumann
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Joël
Ouaknine
Joël Ouaknine
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
ERC grant AVS-ISS (648701) and DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).
James
Worrell
James Worrell
Department of Computer Science, Oxford University, UK
EPSRC Fellowship EP/N008197/1.
10.4230/LIPIcs.ICALP.2021.99
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Eike Neumann, Joël Ouaknine, and James Worrell
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