Identities and periodic oscillations of divide-and-conquer recurrences splitting at half

TL;DR

This paper provides a comprehensive analysis of divide-and-conquer recurrences with splitting at half, revealing a simple identity involving a periodic function and establishing conditions for asymptotic behavior.

Contribution

It introduces a unified identity for solutions of such recurrences, including explicit conditions for the periodic component's properties and broad applicability to various examples.

Findings

Solution always satisfies a specific identity involving a periodic function

Explicit conditions for the continuity and smoothness of the periodic function

Method applies to many examples from literature and can be extended further

Abstract

We study divide-and-conquer recurrences of the form \begin{equation*} f(n) = \alpha f(\lfloor \tfrac n2\rfloor) + \beta f(\lceil \tfrac n2\rceil) + g(n) \qquad(n\ge2), \end{equation*} with $g(n)$ and $f(1)$ given, where $\alpha,\beta\ge0$ with $\alpha+\beta>0$; such recurrences appear often in analysis of computer algorithms, numeration systems, combinatorial sequences, and related areas. We show that the solution satisfies always the simple \emph{identity} \begin{equation*} f(n) = n^{\log_2(\alpha+\beta)} P(\log_2n) - Q(n) \end{equation*} under an optimum (iff) condition on $g(n)$. This form is not only an identity but also an asymptotic expansion because $Q(n)$ is of a smaller order. Explicit forms for the \emph{continuity} of the periodic function $P$ are provided, together with a few other smoothness properties. We show how our results can be easily applied to many dozens of concrete examples collected from the literature, and how they can be extended in various directions. Our method of proof is surprisingly simple and elementary, but leads to the strongest types of results for all examples to which our theory applies.