The Membership Problem for Hypergeometric Sequences with Rational Parameters

Abstract

We investigate the Membership Problem for hypergeometric sequences: given a hypergeometric sequence $⟨u_n{⟩}_{n=0}^{\infty }$ 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.