Another Power Identity Involving Binomial Theorem and Faulhaber's Formula

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Abstract

In this paper, we derive and prove, by means of Binomial theorem and Faulhaber's formula, the following identity between $m$-order polynomials in $T$ $$\sum_{k=1}^{\ell}\sum_{j=0}^m A_{m,j}k^j(T-k)^j=\sum_{k=0}^{m}(-1)^{m-k}U_m(\ell,k)\cdot T^k=T^{2m+1}, \ \ell=T\in\mathbb{N}.$$

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europepmc
last seen: 2026-05-19T01:45:01.086888+00:00
unpaywall
last seen: 2026-05-28T02:00:01.590549+00:00
License: CC-BY-4.0