Sums of powers of integers and generalized Stirling numbers of the second kind

2022/11/21 by José L. Cereceda, Cereceda, José L.
Mathematics · #Advanced Mathematical Identities #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2211.11648

openalex publication_date 2022/11/21 · openalex created_date 2022/11/29 · openalex updated_date 2026/07/28

Abstract

By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers S_k(n) = 1^k + 2^k + ⋯ + n^k, we derive a couple of infinite families of explicit formulas for S_k(n). One of the families involves the r-Stirling numbers of the second kind \genfrac\\0ptkj_r, j=0,1,…,k, while the other involves their duals \genfrac\\0ptkj_-r, with both families of formulas being indexed by the non-negative integer r. As a by-product, we obtain three additional formulas for S_k(n) involving the numbers \genfrac\\0ptkj_n+m, \genfrac\\0ptkj_n-m (where m is any given non-negative integer), and \genfrac\\0ptkj_k-j, respectively. Moreover, we provide a formula for the Bernoulli polynomials B_k(x-1) in terms of \genfrac\\0ptkj_x and the harmonic numbers.

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