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# 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)

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

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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