Identities and periodic oscillations of divideandconquer recurrences splitting at half
Abstract
We study divideandconquer 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2022
 DOI:
 10.48550/arXiv.2210.10968
 arXiv:
 arXiv:2210.10968
 Bibcode:
 2022arXiv221010968H
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics;
 68Q25;
 39B12;
 11B37;
 11B83;
 05A15;
 05A16;
 42A16;
 F.2.2;
 G.2.1;
 G.2.3
 EPrint:
 69 pages, 13 figures, 13 tables