Paper Info
Title
The asymptotic analysis of some interpolated nonlinear recurrence relations
Abstract
We study discrete dynamical systems, or recurrence relations, of the general form [EQUATION] with explicitly known series coefficients αk and α1 ≠ 0. We associate with the discrete system an interpolating continuous system Y (t), such that Y (n) = yn. The asymptotic behaviour of yn can then be investigated through Y (t). The corresponding continuous system is [EQUATION] where G is called the generator (following Labelle's terminology), and is given by an explicit formula in terms of the recurrence relation. This continuous system may fail to be smooth everywhere but nonetheless may be useful. Analytic solution is only rarely possible. We analyze the equation for Y under assumptions of an asymptotic limit, and show that the asymptotic behaviour can be obtained by reverting a series containing logarithms and powers. We introduce a novel reversion based on the Wright ω function. An application of the theory is made to functional iteration of the Lambert W function and the asymptotic behaviour of the iteration is obtained. The iteration of functions is a central topic in the theory of complex dynamical system, and a sophisticated use of conjugation is only one key tool used there. We show here that Labelle's theory and generator can be used to compute the conjugated mapping of functional iterations to simple non-iterative functions in general. We use the Lambert W function again as an example to illustrate this. We also discuss the curious asymptotic series ln z ~ Σk ≥ 1 W<k>(z). This study uses the truncated generalized series tools available in Maple, particularly the logarithmic-and-power series that is usual in Maple. We also use Levin's u-transform as a key piece in interpolating the discrete dynamical system.
Year
DOI
Venue
2014
10.1145/2608628.2608677
ISSAC
Keywords
Field
DocType
algorithms,special function approximations,theory,symbolic integration
Discrete mathematics,Applied mathematics,Recurrence relation,Mathematical analysis,Lambert W function,Asymptotic expansion,Dynamical systems theory,Holonomic function,Asymptotic analysis,Dynamical system,Mathematics,Discrete system
Conference
Citations 
PageRank 
References 
0
0.34
4
Authors
3
Name
Order
Citations
PageRank
Robert M. Corless114321.54
David J. Jeffrey21172132.12
Fei Wang324151.35