On sums of powers of natural numbers

2024/11/02 by Eteri Samsonadze, Samsonadze, Eteri
Computer Science · Mathematics · #05A10 #05A19 #11B68 #40G99 #Advanced Mathematical Identities #Computability, Logic, AI Algorithms #FOS: Mathematics #General Mathematics (math.GM) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2411.11859

openalex publication_date 2024/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The problem of finding the sum of a polynomial's values is considered. In particular, for any n≥ 3, the explicit formula for the sum of the nth powers of natural numbers S_n=∑_x=1^mxⁿ is proved: ∑_x=1^mxⁿ=(-1)ⁿm(m+1)(-(1)/(2)+∑_i=2ⁿa_i(m+2)(m+3)...(m+i)), here a_i=(1)/(i+1)∑_k=1ⁱ\frac(-1)^kkⁿk!(i-k)!, (i=2,3,...,n-1), a_n=((-1)ⁿ)/(n+1). Note that this formula does not contain Bernoulli numbers.

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