Another Power Identity involving Binomial Theorem and Faulhaber's formula

Kolosov Petro (1)

1 Odessa National I.I.Mechnikov University

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

Mots clés: Faulhaber's formula; Faulhaber's theorem; Binomial Theorem; Binomial coefficient; Binomial distribution; Binomial identities; Power Sums; Finite differences

Domaines: Théorie des nombres [math.NT]; Analyse classique [math.CA]; Systèmes dynamiques [math.DS]; Combinatoire [math.CO]; Mathématiques générales [math.GM]

Kolosov Petro. Another Power Identity involving Binomial Theorem and Faulhaber's formula. 2018. ⟨hal-01283042v12⟩