TitleExperimental mathematics applied to the study of non-linear recurrences
NameHogan, Emilie Ann (author), Zeilberger, Doron (chair), Retakh, Vladimir (internal member), Saks, Michael (internal member), Sloane, Neil (outside member), Rutgers University, Graduate School - New Brunswick,
DescriptionIn this thesis we study three topics within the broad fi eld of nonlinear recurrences. First we will consider global asymptotic stability in rational recurrences. A recurrence is globally asymptotically stable when the sequence it produces converges to an equilibrium solution given any initial conditions. Up to now, this topic has not been studied from an algorithmic perspective. We develop an algorithm that takes as input a rational recurrence relation conjectured to be globally asymptotically stable, and, if it is, outputs a rigorous proof of its stability. We apply this algorithm to many speci c rational recurrences. Secondly, we study a three-parameter family of rational recurrences that produce sequences of integers. We apply two methods to prove the integrality of these sequences. We fi rst show that some of the sequences also satisfy a linear recurrence. In order to establish integrality of the entire family we make use of the Laurent phenomenon. Finally, we develop a new concept that generalizes the notion of a recurrence. Instead of producing a single sequence, we produce in finitely many sequences from one set of initial conditions. We will study two families of this type of generalized recurrences that produce rational numbers when complex numbers are expected. We also observe exponential sequences being produced by some of these generalized recurrences.
NoteIncludes bibliographical references
Noteby Emilie Ann Hogan
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.