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

Efficient Evaluations of Weighted Sums over the Boolean Lattice inspired by conjectures of Berti, Corsi, Maspero, and Ventura
By Shalosh B. Ekhad and Doron Zeilberger (with a postscript by Mark van Hoeij)
.pdf    .tex
[original version   ]

First Written: Feb. 22, 2024
Current version of this web-page (adding Christoph Koutschan's confirmations): March 7, 2024.
Previous version (including exciting development by Mark van Hoeij that completes the (rigorous!) proof of the second (until now still open conjecture)): Feb. 28, 2024.
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.

Added Feb. 27, 2024: One of us (DZ) is offering to donate $100 to the OEIS in honor of the first prover of this
challenge
Update, Feb. 28, 2024: Mark van Hoeij met this challenge! A donation to the OEIS Foundation in Mark van Hoeij's honor has been made (do ^F Hoeij).
See the bottom of the new version of this file: challenge
For a detailed explanation see Mark van Hoeij's postscript
Added March 7, 2024: Christoph Koutschan kindly confirmed all the claims in the paper regarding the fact that the sequence defined by the Maple procedures DxH and DxR are identitical, and that the sequences defined by procedures CxH and CxR are the same. See
Christoph Koutschan's postpostscript (and Mathematical notebook).
Putting everything together, this completes the fully rigorous proof of the Berti-Corsi-Maspero-Ventura conjecture (Eq. (4) in the paper).

Added May 10, 2024: Read the fascinating paper by Berto-Corsi-Maspero-Ventura entitled "Infinitely many isolas of mudulational instability for Stokes wave"

Maple package

BCMV.txt, a Maple package for efficient computations of weighted sums over the Boolean lattice

Sample Input and Output for BCMV.txt

If you want to see a fully rigorous proof of the explicit expression for B_p(x) in the paper
the input gives the output.

If you want to see a fully rigorous proof of the explicit expression for A_p(x) (and hence a rigorous new proof of the (already-proved by [BCMV]) Eq. (3)) in the paper
the input gives the output.

If you want to see a verification of Conjecture (4) in the paper, for p from 2 to 1000
the input gives the output.

If you want to see a verification of Conjecture (4) in the paper, for p from 2 to 2000
the input gives the output.

Persoanl Journal of Shalosh B. Ekhad and Doron Zeilberger
Doron Zeilberger's Home Page