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 [Formula: see text] appear in many areas of applied mathematics, from the analysis of algorithms to the optimization of phylogenetic balance indices. These equations are usually "solved" by means of a Master Theorem that provides a bound for the growing order of xn, but not the solution's explicit expression. In this paper we give a finite explicit expression for this solution, in terms of the binary decomposition of n, when the independent term p(n) is a polynomial in ⌈n/2⌉ and ⌊n/2⌋. As an application, we obtain explicit formulas for several sequences of interest in phylogenetics, combinatorics, and computer science, for which no such formulas were known so far: for instance, for the Total Cophenetic index and the rooted Quar

SUBMITTER: Coronado TM 

PROVIDER: S-EPMC9671444 | biostudies-literature | 2022

REPOSITORIES: biostudies-literature