We investigate the Membership Problem for hypergeometric sequences: given a hypergeometric sequence ⟨u_n⟩ of rational numbers and a rational value t, decide whether t occurs in the sequence. We show decidability of this problem under the assumption that in the defining recurrence f(n) u_{n+1} = g(n) u_n, the roots of the polynomials f and g are all rational numbers. We further show the problem remains decidable if the splitting fields of the polynomials f and g are distinct or if f and g are monic polynomials that both split over a quadratic number field.
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.
This talk is based on works done in collaboration with George Kenison, Amaury Pouly, Mahsa Shirmohammadi and James Worrell.
