The Membership Problem for Hypergeometric Sequences with Rational Parameters

02/15/2022

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by   Klara Nosan, et al.

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We investigate the Membership Problem for hypergeometric sequences: given a hypergeometric sequence ⟨ u_n ⟩_n=0^∞ of rational numbers and a target t ∈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.