The Threshold Problem for Hypergeometric Sequences with Quadratic Parameters

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 u_0∈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∈ℕ_0. 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.