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

Abstract

We study divide-and-conquer recurrences of the formf(n)=αf(⌊n2⌋)+βf(⌈n2⌉)+g(n)(n⩾2), with g(n) and f(1) given, where α,β⩾0 with α+β>0; such recurrences appear often in the analysis of computer algorithms, numeration systems, combinatorial sequences, and related areas. We show under an optimum (iff) condition on g(n) that the solution f always satisfies a simple identityf(n)=nlog2⁡(α+β)P(log2⁡n)−Q(n), where P is a periodic function and Q(n) is of a smaller order than the dominant term. This form is thus not only an identity but also an asymptotic expansion. Explicit forms for the 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.