Figurate numbers and sums of powers of integers

TL;DR

This paper proves Marko and Litvinov's conjecture relating powers of integers to hyper-tetrahedron numbers and figurate numbers, establishing explicit formulas and connections with Stirling numbers of the second kind.

Contribution

It confirms the ML conjecture for all natural numbers p and derives new formulas expressing sums of powers as linear combinations of figurate numbers.

Findings

ML conjecture is true for all p

Explicit formulas for sums of powers as figurate numbers

Connection between coefficients and Stirling numbers

Abstract

Recently, Marko and Litvinov (ML) conjectured that, for all positive integers $n$ and $p$, the $p$-th power of $n$ admits the representation $n^p = \sum_{\ell =0}^{p-1} (-1)^{l} c_{p,\ell} F_{n}^{p-\ell}$, where $F_{n}^{p-\ell}$ is the $n$-th hyper-tetrahedron number of dimension $p-\ell$ and $c_{p,\ell}$ denotes the number of $(p -\ell)$-dimensional facets formed by cutting the $p$-dimensional cube $0 \leq x_1, x_2, \ldots, x_p \leq n-1$. 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 $c_{p,\ell} = (p-\ell)! S(p, p-\ell)$, where $S(p,p-\ell)$ are the Stirling numbers of the second kind. Furthermore, we provide a number of equivalent formulas expressing the sum of powers $\sum_{i=1}^{n} i^p$ as a linear combination of figurate numbers.