论文标题
具有有理参数的超几何序列的成员资格问题
The Membership Problem for Hypergeometric Sequences with Rational Parameters
论文作者
论文摘要
我们研究了超几何序列的成员资格问题：给定超测定序列$ \ langle u_n \ rangle_ {n = 0}^\ int \ mathbb {q} $ in \ mathbb {q} $的目标$ t \ in \ mathbb {q} $，确定$ t $是否出现在序列中。我们在定义复发$ p（n）u_ {n} = q（n）u_ {n-1} $的假设中显示了此问题的可决定性，polyenmials $ p（x）$和$ q（x）$的根是所有有理数的。我们的证明依赖于算术进程中的素数密度。我们还观察到了成员问题（和变体）的可决定性与超越理论中的Rohrlich-lang猜想之间的关系。
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.
