Efficient Evaluations of Weighted Sums over the Boolean Lattice inspired by conjectures of Berti, Corsi, Maspero, and Ventura

TL;DR

This paper develops a method to evaluate weighted sums over the Boolean lattice, providing a new proof for one conjecture and supporting evidence for another, which was later proven by others.

Contribution

It introduces a technique to generate and evaluate weighted sums over the Boolean lattice, aiding in the proof of conjectures related to water wave studies.

Findings

New proof of the first conjecture

Overwhelming evidence for the second conjecture

Subsequent proof completed by others

Abstract

In their study of water waves, Massimiliano Berti, Livia Corsi, Alberto Maspero, and Paulo Ventura, came up with two intriguing conjectured identities involving certain weighted sums over the Boolean lattice. They were able to prove the first one, while the second is still open. In this methodological note, we will describe how to generate many terms of these types of weighted sums, and if in luck, evaluate them in closed-form. We were able to use this approach to give a new proof of their first conjecture, and while we failed to prove the second conjecture, we give overwhelming evidence for its veracity. In this second version, we are happy to announce that Mark van Hoeij was able to complete the proof of the second conjecture, by explicitly solving the second-order recurrence mentioned at the end.