  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.
