dc.contributor.author Noble, Rob dc.date.accessioned 2011-03-30T12:01:12Z dc.date.available 2011-03-30T12:01:12Z dc.date.issued 2011-03-30 dc.identifier.uri http://hdl.handle.net/10222/13298 dc.description.abstract In this thesis we study the zeros and asymptotics of sequences that satisfy linear recurrence relations with generally nonconstant coefficients. en_US By the theorem of Skolem-Mahler-Lech, the set of zero terms of a sequence that satisfies a linear recurrence relation with constant coefficients taken from a field of characteristic zero is comprised of the union of finitely many arithmetic progressions together with a finite exceptional set. Further, in the nondegenerate case, we can eliminate the possibility of arithmetic progressions and conclude that there are only finitely many zero terms. For generally nonconstant coefficients, there are generalizations of this theorem due to Bézivin and to Methfessel that imply, under fairly general conditions, that we obtain a finite union of arithmetic progressions together with an exceptional set of density zero. Further, a condition is given under which one can exclude the possibility of arithmetic progressions and obtain a set of zero terms of density zero. In this thesis, it is shown that this condition reduces to the nondegeneracy condition in the case of constant coefficients. This allows for a consistent definition of nondegeneracy valid for generally nonconstant coefficients and a unified result is obtained. The asymptotic theory of sequences that satisfy linear recurrence relations with generally nonconstant coefficients begins with the basic theorems of Poincaré and Perron. There are some generalizations of these theorems that hold in greater generality, but if we restrict the coefficient sequences of our linear recurrences to be polynomials in the index, we obtain full asymptotic expansions of a predictable form for the solution sequences. These expansions can be obtained by applying a transfer method of Flajolet and Sedgewick or, in some cases, by applying a bivariate method of Pemantle and Wilson. In this thesis, these methods are applied to a family of binomial sums and full asymptotic expansions are obtained. The leading terms of the expansions are obtained explicitly in all cases, while in some cases a field containing the asymptotic coefficients is obtained and some divisibility properties for the asymptotic coefficients are obtained using a generalization of a method of Stoll and Haible. dc.language.iso en en_US dc.subject Skolem-Mahler-Lech, nondegenerate linear recurrence sequences, recurrence sequences with nonconstant coefficients, asymptotics, binomial sums, multivariate sequences, generalized Riordan arrays, central Delannoy numbers en_US dc.title Zeros and Asymptotics of Holonomic Sequences en_US dc.date.defence 2011-03-21 dc.contributor.department Department of Mathematics & Statistics - Math Division en_US dc.contributor.degree Doctor of Philosophy en_US dc.contributor.external-examiner Dr. Peter Paule en_US dc.contributor.graduate-coordinator Dr. Jason Brown en_US dc.contributor.thesis-reader Dr. Keith Johnson en_US dc.contributor.thesis-reader Dr. Sara Faridi en_US dc.contributor.thesis-supervisor Dr. Karl Dilcher en_US dc.contributor.ethics-approval Not Applicable en_US dc.contributor.manuscripts Not Applicable en_US dc.contributor.copyright-release Yes en_US
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