$N$单位根Riesz能量的显式公式
Explicit formulas for the Riesz energy of the $N$th roots of unity
摘要：
本文Brauchart，Hardin和Saff [Bull. Lond. Math.Soc.41（4）（2009）]给出了单位根$N$的Riesz $s$ -能量的完全渐近展开式，其在Cohn和Kumar的意义下形成单位圆上的点的普遍最优分布[J. Amer.Math.Soc.20（2007）]。这里，对于当$s$是偶数的情况，获得精确公式（对所有$N \geq 2$有效）。在奇异Riesz $s$ -势$1/r^s$，$r$为两点间的欧几里得距离的情况下，使用Riesz能量的连续修正能量近似.第一类Stirling数、欧拉数和部分Bell多项式的特殊值起着核心作用。这些数量之间的几个身份显示。
英文摘要：
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.
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