A sequence is holonomic if its terms obey a linear recurrence relation with polynomial coefficients. In this paper we consider the Positivity Problem for second-order holonomic sequences with linear coefficients, i.e., the question of determining, for a given sequence (un)n obeying the recurrence (a1n+a0)un = (b1n+b0)un-1 + (c1n+c0)un-2, whether all terms of (un)n are non-negative. Our main result establishes decidability in the case of two distinct rational characteristic roots. We achieve this by leveraging recent results on effective transcendence of values of E-functions and 1-periods, which are integrals playing a central role in the theory of algebraic curves.
Submitted, 2026. 20 pages.
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© 2026 George Kenison, Oleksiy Klurman, Engel
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