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Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\toK$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in$K[z]$. Given that $P_d(z)$ is a nonzero constant polynomial, we show that theset of $n\in \mathbb{N}$ for which $f(n)=0$ is a union of finitely manyarithmetic progressions and a finite set. This generalizes theSkolem-Mahler-Lech theorem, which assumes that $f(n)$ satisfies a linearrecurrence. We discuss examples and connections to the set of zero coefficientsof a power series satisfying a homogeneous linear differential equation withrational function coefficients.

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