Exercises in Iterational Asymptotics

TL;DR

This work investigates the fine-grained asymptotics of nonlinear iterates by extracting leading exponential corrections and subleading logarithmic terms via product representations and asymptotic expansions. It develops a general framework for constants $C$ (and $C(q)$) that capture long-run behavior across several recurrences, providing explicit high-order expansions for cases such as $x_k = p x_{k-1}(1 \pm x_{k-1})$ and $x_k = x_{k-1} + 1/x_{k-1}^{q}$, including leading log corrections and numerically estimated constants. The paper introduces a brute-force coefficient-matching approach to obtain higher-order terms, reveals connections to sequences and number-theoretic phenomena, and uncovers reciprocity relations that yield additional constants (e.g., $\Lambda$) for transformed recurrences. These results advance the practical analysis of iterative maps by delivering precise asymptotics and highlighting deeper structural patterns, with potential applications to broader classes of nonlinear recurrences.

Abstract

The problems and solutions contained here, all associated with nonlinear recurrences and long-term trends, are new (as far as is known).