A remark on exact formulas for the Riesz energy of the $N$th roots of unity

Details A remark on exact formulas for the Riesz energy of the $N$th roots of unity A remark on exact formulas for the Riesz energy of the $N$th roots of unity

2011-05-27

arxiv.org/abs/1105.5530v1

Physics

Mathematical Physics

18 pages

Scientific paper

In the paper the author, Hardin and Saff [Bull. Lond. Math. Soc. 41 (2009), no. 4] we provided the complete asymptotic expansions of the Riesz $s$-energy of the $N$th roots of unity. In this note the exact formulas (valid for all $N \geq 2$) are obtained for $s$ is an even integer. Explicit representations of the coefficients of the powers of $N$ are derived by direct computations. In the case of positive $s$ 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. A number of identities between these quantities are shown.

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