Explicit formulas for the Riesz energy of the $N$th roots of unity

TL;DR

This paper derives explicit formulas for the Riesz energy of the Nth roots of unity when the parameter s is an even integer, using combinatorial numbers and special polynomials, extending previous asymptotic results.

Contribution

It provides exact formulas for the Riesz energy of roots of unity for all N when s is an even integer, involving Stirling, Eulerian, and Bell numbers.

Findings

Exact formulas valid for all N when s is an even integer.

Connections established between combinatorial numbers and Riesz energy.

New identities among Stirling, Eulerian, and Bell numbers.

Abstract

The paper Brauchart, Hardin and Saff [Bull. Lond. Math. Soc. 41(4) (2009)] gives the complete asymptotic expansions of the Riesz $s$-energy of the $N$th roots of unity which form a universally optimal distribution of points on the unit circle in the sense of Cohn and Kumar [J. Amer. Math. Soc. 20 (2007)]. Here, exact formulas (valid for all $N \geq 2$) are obtained for the case when $s$ is an even integer. In the case of the singular Riesz $s$-potential $1/r^s$, $r$ the Euclidean distance between two points, a continuous modified energy approximation of the Riesz energy is used. Stirling numbers of the first kind, Eulerian numbers and special values of partial Bell polynomials play a central role. Several identities between these quantities are shown.