On the sum of k-th powers in terms of earlier sums

2019/12/16 by Steven J. Miller, Miller, Steven J., Enrique Treviño +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Identities #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.1912.07171

Abstract

For k a positive integer let S_k(n) = 1^k + 2^k + ⋯ + n^k, i.e., S_k(n) is the sum of the first k-th powers. Faulhaber conjectured (later proved by Jacobi) that for k odd, S_k(n) could be written as a polynomial of S₁(n); for example S₃(n) = S₁(n)². We extend this result and prove that for any k there is a polynomial g_k(x,y) such that S_k(n) = g(S₁(n), S₂(n)). The proof yields a recursive formula to evaluate S_k(n) as a polynomial of n that has roughly half the number of terms as the classical one.

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