Hypergeometric sequences are rational-valued sequences that satisfy first-order linear recurrence relations with polynomial coefficients; that is, a hypergeometric sequence $\langle u_n \rangle_{n=0}^{\infty}$ is one that satisfies a recurrence of the form $f(n)u_n = g(n)u_{n-1}$ where $f,g \in \mathbb{Z}[x]$. In this paper, we consider the Membership Problem for hypergeometric sequences: given a hypergeometric sequence $\langle u_n \rangle_{n=0}^{\infty}$ and a target value $t\in \mathbb{Q}$, determine whether $u_n=t$ for some index $n$. We establish decidability of the Membership Problem under the assumption that either (i) $f$ and $g$ have distinct splitting fields or (ii) $f$ and $g$ are monic polynomials that both split over a quadratic extension of $\mathbb{Q}$. Our results are based on an analysis of the prime divisors of polynomial sequences $\langle f(n) \rangle_{n=1}^\infty$ and $\langle g(n) \rangle_{n=1}^\infty$ appearing in the recurrence relation.

翻译：超几何序列是满足具有多项式系数的一阶线性递推关系的有理数值序列；即，超几何序列$\langle u_n \rangle_{n=0}^{\infty}$满足形如$f(n)u_n = g(n)u_{n-1}$的递推关系，其中$f,g \in \mathbb{Z}[x]$。本文研究了超几何序列的成员问题：给定一个超几何序列$\langle u_n \rangle_{n=0}^{\infty}$和一个目标值$t\in \mathbb{Q}$，判断是否存在索引$n$使得$u_n=t$。我们在以下假设下建立了成员问题的可判定性：（i）$f$和$g$具有不同的分裂域，或（ii）$f$和$g$是首一多项式，且均在$\mathbb{Q}$的二次扩张上分裂。我们的结果基于对递推关系中出现的多项式序列$\langle f(n) \rangle_{n=1}^\infty$和$\langle g(n) \rangle_{n=1}^\infty$的素因子的分析。