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

Published online by Cambridge University Press:  22 February 2012

Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC, V5A 1S6Canada. e-mail:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada. e-mail:
Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC, V5A 1S6, Canada. e-mail:


Let 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.

Research Article
Copyright © Cambridge Philosophical Society 2012

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