A matrix variation on Ramus's identity for lacunary sums of binomial coefficients

TL;DR

This paper explores a matrix-based variation of Ramus's identity for lacunary binomial sums, linking it to harmonic numbers related to Fermat's Last Theorem, extending classical results into matrix algebra.

Contribution

It introduces a novel matrix variation of Ramus's identity, generalizing the classical scalar form to matrices of arbitrary dimension.

Findings

Matrix variation of Ramus's identity derived

Connection established between lacunary sums and matrix harmonic numbers

Potential implications for number theory and algebraic identities

Abstract

We study the well-known lacunary sums of binomial coefficients considered, most notably, by Christian Ramus, and their connection to a special kind of harmonic number associated with the first case of Fermat's Last Theorem. For one case of Ramus's famous identity we obtain a variation in which some of the parameters are replaced by square matrices of arbitrary dimension.