The Membership Problem for Hypergeometric Sequences with Rational Parameters

TL;DR

This paper proves the decidability of the Membership Problem for certain hypergeometric sequences with rational parameters, linking it to prime density bounds and a conjecture in transcendence theory.

Contribution

It establishes decidability results for hypergeometric sequences with rational roots, connecting the problem to prime distribution and transcendence conjectures.

Findings

Decidability of membership for hypergeometric sequences with rational roots.

Connection between the problem and prime density in arithmetic progressions.

Relation to the Rohrlich-Lang conjecture in transcendence theory.

Abstract

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.