An unusual identity for odd-powers

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.

4 pages

References in corpus (1)

• Formulas for sums of powers of integers and their reciprocals