Another Power Identity involving Binomial Theorem and Faulhaber's formula

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

\(P_m(\ell,T) := \sum_{k=1}^{\ell}\sum_{j=0}^m A_{m,j}k^j(T-k)^j=\sum_{k=0}^{m}(-1)^{m-k}U_m(\ell,k)\cdot T^k\)

arXiv version: https://arxiv.org/abs/1603.02468

Related OEIS sequences: A287326, A300656, A300785, A302971, A304042, A316349, A316387

Keywords: Faulhaber's formula, Faulhaber's theorem, Binomial Theorem, Binomial coefficient, Binomial distribution, Binomial identities, Power Sums, Finite differences

MSC classes: 11C08, 41A10

See also:

https://kolosovpetro.github.io/research/binomial-theorem-and-convolution/