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# The Threshold Problem for Hypergeometric Sequences with Quadratic Parameters

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## Acknowledgements

I am grateful to both Mahsa Shirmohammadi and James Worrell for many helpful discussions. I also thank the anonymous reviewers for their constructive feedback.

## Cite As

George Kenison. The Threshold Problem for Hypergeometric Sequences with Quadratic Parameters. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 145:1-145:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.145

## Abstract

Hypergeometric sequences are rational-valued sequences that satisfy first-order linear recurrence relations with polynomial coefficients; that is, ⟨u_n⟩_{n=0}^∞ is hypergeometric if it satisfies a first-order linear recurrence of the form p(n)u_{n+1} = q(n)u_n with polynomial coefficients p,q ∈ ℤ[x] and u₀ ∈ ℚ. In this paper, we consider the Threshold Problem for hypergeometric sequences: given a hypergeometric sequence ⟨u_n⟩_{n=0}^∞ and a threshold t ∈ ℚ, determine whether u_n ≥ t for each n ∈ ℕ₀. We establish decidability for the Threshold Problem under the assumption that the coefficients p and q are monic polynomials whose roots lie in an imaginary quadratic extension of ℚ. We also establish conditional decidability results; for example, under the assumption that the coefficients p and q are monic polynomials whose roots lie in any number of quadratic extensions of ℚ, the Threshold Problem is decidable subject to the truth of Schanuel’s conjecture. Finally, we show how our approach both recovers and extends some of the recent decidability results on the Membership Problem for hypergeometric sequences with quadratic parameters.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Discrete mathematics
• Computing methodologies → Symbolic and algebraic algorithms
• Computing methodologies → Algebraic algorithms
##### Keywords
• Threshold Problem
• Membership Problem
• Hypergeometric Sequences

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