# Multidimensional Catalan and related numbers as Hausdorff moments

Source: [Multidimensional Catalan and related numbers as Hausdorff moments](<https://www.sota2.com/research/paper/68fa509c1e209131ce96cb0d>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T10:56:28.899Z

Authors: K. Gorska, K. A. Penson

Published: 2013-04-22T16:28:29.000Z

## Abstract

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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1304.6008>)

[PDF](<https://arxiv.org/pdf/1304.6008.pdf>)