Some Extensions of the Skolem-Mahler-Lech Theorem

97 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1983.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.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD