Multidimensional Catalan and related numbers as Hausdorff moments

Abstract

We study integral representation of so-called $d$-dimensional Catalan numbers $C_d(n)$, defined by $[{\prod }_{p=0}^{d-1}\frac{p!}{(n+p)!}](dn)!$, $d=2,3,...$, $n=0,1,...$. 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 $D_d(n)=[{\prod }_{p=0}^{d-1}\frac{p!}{(n+p)!}][{\prod }_{q=0}^{d/2-1}\frac{(2n+2q)!}{(2q)!}]$ is analyzed along the same lines.

comments added, two new references added

References in corpus (2)

• Product of Ginibre matrices: Fuss-Catalan and Raney distributions
• Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices

Cited by in corpus (3)

• Noncrossing sets and a Graßmann associahedron
• Spin Multiplicities
• Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution