$N$维中的马德隆常数
The Madelung Constant in $N$ Dimensions
摘要：
我们介绍了两个收敛级数展开（直接和递归）的贝塞尔函数和表示的总和$r_N(m)$的平方$N$维马德隆常数，$M_N(s)$，其中$s$是指数的马德隆系列（通常选择为$s=1/2$）。详细讨论了函数的性质，包括解析延拓和Bessel函数展开的收敛性。递归定义用于评估$r_N(m)$。给出了$s=\tfrac{1}{2}, \tfrac{3}{2}, 3$的$M_N(s)$值和$N=20$以下尺寸的$M_N(s)$值以及$M_N(1/2)$至$N=100$的$M_N(s)$值。本文简要分析了Zucker关于$N=8$维马德隆常数的原始分析及其在高维中的可能延续。
英文摘要：
We introduce two convergent series expansions (direct and recursive) in terms of Bessel functions and representations of sums $r_N(m)$ of squares for $N$-dimensional Madelung constants, $M_N(s)$, where $s$ is the exponent of the Madelung series (usually chosen as $s=1/2$). The functional behavior including analytical continuation, and the convergence of the Bessel function expansion is discussed in detail. Recursive definitions are used to evaluate $r_N(m)$. Values for $M_N(s)$ for $s=\tfrac{1}{2}, \tfrac{3}{2}, 3$ and 6 for dimension up to $N=20$ and for $M_N(1/2)$ up to $N=100$ are presented. Zucker's original analysis on $N$-dimensional Madelung constants for even dimensions up to $N=8$ and their possible continuation into higher dimensions is briefly analyzed.
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