On the set of zero coefficients of a function satisfying a linear differential equation

Abstract

AbstractLet K be a field of characteristic zero and suppose that f: → K satisfies a recurrence of the form \[ f(n) = \sum_{i=1}^d P_i(n) f(n-i), \] for n sufficiently large, where P1(z),. . .,Pd(z) are polynomials in K[z]. Given that Pd(z) is a nonzero constant polynomial, we show that the set of n ∈ for which f(n) = 0 is a union of finitely many arithmetic progressions and a finite set. This generalizes the Skolem–Mahler–Lech theorem, which assumes that f(n) satisfies a linear recurrence. We discuss examples and connections to the set of zero coefficients of a power series satisfying a homogeneous linear differential equation with rational function coefficients.