Preprint Article Version 2 This version is not peer-reviewed

# Another Power Identity Involving Binomial Theorem and Faulhaber's Formula

Version 1 : Received: 19 October 2018 / Approved: 19 October 2018 / Online: 19 October 2018 (10:35:13 CEST)
Version 2 : Received: 21 October 2018 / Approved: 22 October 2018 / Online: 22 October 2018 (11:02:08 CEST)

How to cite: Kolosov, P. Another Power Identity Involving Binomial Theorem and Faulhaber's Formula. Preprints 2018, 2018100446 (doi: 10.20944/preprints201810.0446.v2). Kolosov, P. Another Power Identity Involving Binomial Theorem and Faulhaber's Formula. Preprints 2018, 2018100446 (doi: 10.20944/preprints201810.0446.v2).

## 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 $T$ $$\sum_{k=1}^{\ell}\sum_{j=0}^m A_{m,j}k^j(T-k)^j=\sum_{k=0}^{m}(-1)^{m-k}U_m(\ell,k)\cdot T^k=T^{2m+1}, \ \ell=T\in\mathbb{N}.$$

## Subject Areas

Faulhaber's formula; Faulhaber's theorem; binomial theorem; binomial coefficient; binomial distribution; binomial identities; power sums; finite differences