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https://hdl.handle.net/2142/71216
Description
Title
Some Extensions of the Skolem-Mahler-Lech Theorem
Author(s)
Laohakosol, Vichian
Issue Date
1983
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
The Skolem-Mahler-Lech theorem states that if the Taylor series expansion (about the origin) of a rational function has infinitely many zero coefficients, then the set of indices of these zero coefficients forms a finite union of arithmetic progressions modulo a finite set. Since rational functions satisfy linear differential equations of order 0 with polynomial coefficients, it is natural to conjecture that the Skolem-Mahler-Lech theorem also holds for functions satisfying linear differential equations with polynomial coefficients of higher orders. In this thesis, we treat the case of first order differential equations. We are able to answer affirmatively parts of this conjecture, for example, we are able to show that functions satisfying first order linear homogeneous differential equations with polynomial coefficients of Fuchsian type are skomal. We also extend the Skolem-Mahler-Lech theorem in some other directions from the case of rational functions, and obtain certain arithmetical properties for functions satisfying linear differential equations with polynomial coefficients.
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