Multidimensional Catalan and related numbers as Hausdorff moments

TL;DR

This paper explores the integral representation of multidimensional Catalan numbers as Hausdorff moments, providing explicit forms of the associated functions and analyzing their properties, with implications for combinatorics and mathematical analysis.

Contribution

It introduces explicit forms of functions representing multidimensional Catalan numbers as Hausdorff moments, extending the understanding of their integral representations and uniqueness.

Findings

Proved that $C_{d}(n)$ are Hausdorff moments of positive functions $W_{d}(x)$.

Constructed explicit forms of $W_{d}(x)$ using hypergeometric functions.

Analyzed the properties and graphical representations of these functions.

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.