%Efficient Evaluations of Weighted Sums over the Boolean Lattice inspired by conjectures of Berti, Corsi, Maspero, and Ventura
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{\bf
Efficient Evaluations of Weighted Sums over the Boolean Lattice
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{\bf
inspired by conjectures of Berti, Corsi, Maspero, and Ventura
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{\it Shalosh B. EKHAD and Doron ZEILBERGER}
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{\bf 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.
{\bf An Intriguing Email message from Alberto Maspero}
Awhile ago one of us (DZ) received an email message [M],
with the following.
\quad \quad {\it In our current study of water waves [BCMV], continuing our work in [BMV], and other papers, we came across the following sums.}
Let $p \geq 2$, $1 \leq q \leq p-1$ and $0