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 $p(n)u_{n}=q(n)u_{n-1}$, 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 $t\ mathb ⁇ $, 我们根据以下假设来判断这一问题是否在序列中发生。 在定义重现$p(n)u ⁇ n ⁇ q(n) ⁇ u ⁇ u ⁇ n-1}$时, 多数值的根值$p(x)$和$q(x)$都是合理的数字。 我们的证据依赖于计算进程中质数密度的界限。 我们还观察了会籍问题(和变体)的衰减性和超越理论的Rohrlich-Langjecture之间的关系 。
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