Explicit solution of divide-and-conquer dividing by a half recurrences with polynomial independent term

Abstract

Divide-and-conquer dividing by a half recurrences, of the form $x_n =a\cdot x_{\left\lceil{n}/{2}\right\rceil}+a\cdot x_{\left\lfloor{n}/{2}\right\rfloor}+p(n)$, $n\geq 2$, appear in many areas of applied mathematics, from the analysis of algorithms to the optimization of phylogenetic balance indices. The Master Theorems that solve these equations do not provide the solution's explicit expression, only its big-$\Theta$ order of growth. In this paper we give an explicit expression (in terms of the binary decomposition of $n$) for the solution $x_n$ of a recurrence of this form, with given initial condition $x_1$, when the independent term $p(n)$ is a polynomial in $\lceil{n}/{2}\rceil$ and $\lfloor{n}/{2}\rfloor$.