  We study integral representation of so-called $d$-dimensional Catalan numbers
$C_{d}(n)$, defined by $[\prod_{p=0}^{d-1} \frac{p!}{(n+p)!}] (d n)!$, $d = 2,
3, ...$, $n=0, 1, ...$. We prove that the $C_{d}(n)$'s are the $n$th Hausdorff
power moments of positive functions $W_{d}(x)$ defined on $x\in[0, d^d]$. We
construct exact and explicit forms of $W_{d}(x)$ and demonstrate that they can
be expressed as combinations of $d-1$ hypergeometric functions of type
$_{d-1}F_{d-2}$ of argument $x/d^d$. These solutions are unique. We analyse
them analytically and graphically. A combinatorially relevant, specific
extension of $C_{d}(n)$ for $d$ even in the form $D_{d}(n)=[\prod_{p = 0}^{d-1}
\frac{p!}{(n+p)!}] [\prod_{q = 0}^{d/2 - 1} \frac{(2 n + 2 q)!}{(2 q)!}]$ is
analyzed along the same lines.
