An unusual identity for odd-powers

TL;DR

This paper introduces a novel polynomial pattern that expresses odd powers as a double sum involving binomial-like coefficients, offering a new perspective on polynomial expansions.

Contribution

It presents a new polynomial expansion pattern for odd powers, expanding the understanding of polynomial identities and their representations.

Findings

Derived a new polynomial expansion formula for odd powers.

Provided explicit coefficients for the polynomial pattern.

Potential applications in algebraic identities and polynomial analysis.

Abstract

In this manuscript we provide a new polynomial pattern. This pattern allows to find a polynomial expansion of the form \[x^{2m+1} = \sum_{k=1}^{x}\sum_{r=0}^{m} \mathbf{A}_{m,r} k^r (x-k)^r,\] where $x,m\in\mathbb{N}$ and $\mathbf{A}_{m,r}$ is real coefficient.