Integral values of generating functions of recursive sequences

Abstract

Suppose that a0,a1,… is an integer sequence which satisfies a recurrence relation with constant coefficients, and let T(x)=f(x)/g(x) be its generating function, where f(x) and g(x) have no common factors in Z[x]. In this article, we study the problem of finding the rational values of x such that T(x) is an integer. We say that such a number is good for the sequence. Our first main result is that if g(x) has at least two different irreducible factors, or if g(x) has a single irreducible factor of degree at least 3, then the sequence has only finitely many good values. We also study sequences of the form 0,1,… for which the recurrence relation has order 2. Among other results, we show that under a mild condition on the recurrence relation, the sequence has infinitely many good values, and we give a constructive method to find all of them.