Figurate numbers and sums of powers of integers

2020/01/09 by José L. Cereceda, Cereceda, José L.
Mathematics · Physics and Astronomy · #11B83 (Primary) 11B57 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2001.03208

openalex publication_date 2020/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Recently, Marko and Litvinov (ML) conjectured that, for all positive integers n and p, the p-th power of n admits the representation n^p = ∑_{ℓ =0}^p-1 (-1)^l c_p,ℓ F_n^p-ℓ, where F_n^p-ℓ is the n-th hyper-tetrahedron number of dimension p-ℓ and c_p,ℓ denotes the number of (p -ℓ)-dimensional facets formed by cutting the p-dimensional cube 0 ≤ x₁, x₂, …, x_p ≤ n-1. In this paper we show that the ML conjecture is true for every natural number p. Our proof relies on the fact that the validity of the ML conjecture necessarily implies that c_p,ℓ = (p-ℓ)! S(p, p-ℓ), where S(p,p-ℓ) are the Stirling numbers of the second kind. Furthermore, we provide a number of equivalent formulas expressing the sum of powers ∑_i=1ⁿ i^p as a linear combination of figurate numbers.

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