Hypergeometric sequences are rational-valued sequences that satisfy first-order linear recurrence relations with polynomial coefficients; that is, $\langle u_n \rangle_{n=0}^\infty$ 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 $p,q\in\mathbb{Z}[x]$ and $u_0\in\mathbb{Q}$. In this paper, we consider the Threshold Problem for hypergeometric sequences: given a hypergeometric sequence $\langle u_n\rangle_{n=0}^\infty$ and a threshold $t\in\mathbb{Q}$, determine whether $u_n \ge t$ for each $n\in\mathbb{N}_0$. 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 $\mathbb{Q}$. 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 $\mathbb{Q}$, 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.

翻译：超几何序列是满足一阶线性递推关系（系数为多项式）的有理数值序列；即，若序列$\langle u_n \rangle_{n=0}^\infty$满足系数$p,q\in\mathbb{Z}[x]$和$u_0\in\mathbb{Q}$的一阶线性递推$p(n)u_{n+1} = q(n)u_{n}$，则其为超几何序列。本文研究超几何序列的阈值问题：给定超几何序列$\langle u_n\rangle_{n=0}^\infty$和阈值$t\in\mathbb{Q}$，判定是否对所有$n\in\mathbb{N}_0$均有$u_n \ge t$。我们在系数$p$和$q$为首一多项式且其根位于$\mathbb{Q}$的虚二次扩域的假设下，建立了阈值问题的可判定性。我们还获得了条件性可判定结果；例如，在系数$p$和$q$为首一多项式且其根位于$\mathbb{Q}$的任意多个二次扩域的假设下，若Schanuel猜想成立，则阈值问题可判定。最后，我们展示了该方法如何恢复并扩展了近期关于二次参数超几何序列成员资格问题的若干可判定性结果。