We investigate the Membership Problem for hypergeometric sequences: given a hypergeometric sequence $\langle u_n \rangle_{n=0}^\infty$ of rational numbers and a target $t \in \mathbb{Q}$, decide whether $t$ occurs in the sequence. We show decidability of this problem under the assumption that in the defining recurrence $p(n)u_{n+1}=q(n)u_n$, the roots of the polynomials $p(x)$ and $q(x)$ are all rational numbers. Our proof relies on bounds on the density of primes in arithmetic progressions. We also observe a relationship between the decidability of the Membership problem (and variants) and the Rohrlich-Lang conjecture in transcendence theory.

翻译：我们调查了超几何序列的会籍问题:根据超几何序列 $\ langle u_n\rgle ⁇ n=0\ ⁇ infty$ 理性数字和目标$t\ mathbb ⁇ $, 确定是否在序列中发生 $t 。 我们显示了这一问题的可变性, 假设在定义重现 $p(n)u ⁇ n+1q(n)u_n$时, 多数值的根部$p(x)$和$q(x)$都是合理的数字。 我们的证据取决于计算过程中质的密度。 我们还观察了会籍问题( 和变体) 的可变性与超越理论的 Rohrlich- Lang 相近性之间的关系 。