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arXiv · 2508.00944

Positivity of Nearly Linearly Recurrent Sequences

Amaury Pouly·Mahsa Shirmohammadi·James Worrell
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Abstract

Nearly linear recurrences generalise linear recurrences and can be represented as special cases of both linear time-invariant systems in control theory and linear-constraint loops in program analysis. We formulate the Positivity Problem for such recurrences: given a recurrence and initial values, decide whether every sequence satisfying the recurrence is termwise nonnegative. This problem generalises Positivity for linear recurrence sequences and is a special case of halfspace non-reachability for linear time-invariant systems. Our main result is a decision procedure for order-2 recurrences. The termination of the procedure relies on a transcendence theorem of independent interest: we prove that certain convergent series obtained by summing the absolute values of terms of algebraic linear recurrence sequences are transcendental.

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Amaury Pouly, Mahsa Shirmohammadi, James Worrell. 2026-07-26. Positivity of Nearly Linearly Recurrent Sequences. https://arxiv.org/abs/2508.00944

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