Rational series and asymptotic expansion for linear homogeneous divide-and-conquer recurrences

TL;DR

This paper introduces a linear algebra-based method to compute the asymptotic expansion of rational sequences satisfying divide-and-conquer recurrences, bridging a gap in their asymptotic analysis.

Contribution

It presents a novel mechanical process using Jordan normal form and spectral radius to analyze asymptotics of rational divide-and-conquer sequences, complementing existing analytic methods.

Findings

Method effectively computes asymptotic expansions

Comparison with analytic number theory approaches

New techniques for Fourier series of periodic functions

Abstract

Among all sequences that satisfy a divide-and-conquer recurrence, the sequences that are rational with respect to a numeration system are certainly the most immediate and most essential. Nevertheless, until recently they have not been studied from the asymptotic standpoint. We show how a mechanical process permits to compute their asymptotic expansion. It is based on linear algebra, with Jordan normal form, joint spectral radius, and dilation equations. The method is compared with the analytic number theory approach, based on Dirichlet series and residues, and new ways to compute the Fourier series of the periodic functions involved in the expansion are developed. The article comes with an extended bibliography.