Applications of transcendental number theory to decision problems for hypergeometric sequences

A rational-valued sequence is hypergeometric if it satisfies a first-order linear recurrence relation with polynomial coefficients. In this note we discuss two decision problems, the membership and threshold problems, for hypergeometric sequences. The former problem asks whether a chosen target is in the orbit of a given sequence, whilst the latter asks whether every term in a sequence is bounded from below by a given value. We establish decidability results for restricted variants of these two decision problems with an approach via transcendental number theory. Our contributions include the following: the membership and threshold problems are both decidable for the class of rational-valued hypergeometric sequences with Gaussian integer parameters.