A Short Proof of a Concrete Sum

TL;DR

This paper presents a concise proof of a generalized Hermite identity using Fourier expansion and trigonometry, offering an alternative to modular arithmetic methods.

Contribution

It introduces a novel, shorter proof technique for a generalized Hermite sum formula based on Fourier and trigonometric approaches.

Findings

Provides a simpler proof of a generalized Hermite identity

Utilizes Fourier expansion of the floor function

Employs trigonometric formulas for the proof

Abstract

We give an alternative proof of a formula that generalizes Hermite's identity. Instead involving modular arithmetic, our short proof relies on the Fourier-type expansion for the floor function and on a trigonometric formula.