3 research outputs found
On the set of zero coefficients of a function satisfying a linear differential equation
Let be a field of characteristic zero and suppose that satisfies a recurrence of the form
for sufficiently large, where are polynomials in
. Given that is a nonzero constant polynomial, we show that the
set of for which is a union of finitely many
arithmetic progressions and a finite set. This generalizes the
Skolem-Mahler-Lech theorem, which assumes that 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.Comment: 11 page