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 $xin[0, d^d]$. 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.
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