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 α_k and α₁ ≠ 0. We associate with the discrete system an interpolating continuous system Y (t), such that Y (n) = y_n. The asymptotic behaviour of y_n 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 ~ Σ_{k ≥ 1} W(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.