On the set of zero coefficients of a function satisfying a linear differential equation
2012 ◽
Vol 153
(2)
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pp. 235-247
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Keyword(s):
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.
1952 ◽
Vol 48
(3)
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pp. 428-435
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1942 ◽
Vol 46
(378)
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pp. 146-151
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1987 ◽
Vol 101
(2)
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pp. 317-317
1950 ◽
Vol 46
(4)
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pp. 570-580
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2014 ◽
Vol 2
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pp. 173-188