The Membership Problem for Hypergeometric Sequences with Quadratic Parameters

George Kenison, Klara Nosan, Mahsa Shirmohammadi, James Worrell

Hypergeometric sequences are rational-valued sequences that satisfy first-order linear recurrence relations with polynomial coefficients; that is, a hypergeometric sequence ⟨u_n⟩ satisfies a recurrence relation of the form f(n)u_n = g(n)u_{n-1} where f and g are polynomials with integer coefficients.

In this paper, we consider the Membership Problem for hypergeometric sequences: given a hypergeometric sequence ⟨u_n⟩ and a rational value t, determine whether u_n=t for some index n. We establish decidability of the Membership Problem under the assumption that either (i) f and g have distinct splitting fields or (ii) f and g are monic polynomials that both split over a quadratic number field.

Our results are based on an analysis of the prime divisors of polynomials.