Sums of powers of integers via differentiation

2023/03/07 by Cereceda, José L.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2303.04122

Abstract

For integer k ≥ 0, let S_k denote the sum of the kth powers of the first n positive integers 1^k + 2^k + ⋯ + n^k. For any given k, the power sum S_k can in principle be determined by differentiating k times (with respect to x) the associated exponential generating function ∑_k=0^∞S_k x^k/k!, and then taking the limit of the resulting differentiated function as x approaches 0. In this paper, we exploit this method to establish a couple of seemingly novel recurrence relations, one of them involving the even-indexed power sums S₂, S₄,…, S_2k, and the other the odd-indexed power sums S₁, S₃, …, S_2k-1, with both recurrence relations depending explicitly on the parameter N = n + (1)/(2). From this, we obtain a determinantal formula of order k which yields S_2k [S_2k-1] in the Faulhaber form, that is, as an odd [even] polynomial in N. As a byproduct, we discover a new determinantal formula for the Bernoulli number B_2k. Furthermore, we show that S_2k and S_2k-1 can be obtained by taking the corresponding higher order derivatives of the Chebyshev polynomials of the second kind.

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