Multidimensional Catalan and related numbers as Hausdorff moments

الأصل · EN

We study integral representation of so-called d-dimensional Catalan numbers Cd(n), defined by [∏ₚ₌₀ᵈ⁻¹ p!/(n+p)!] (d n)!, d = 2, 3,..., n=0, 1,.... We prove that the Cd(n)'s are the nth Hausdorff power moments of positive functions Wd(x) defined on x∈[0, dᵈ]. We construct exact and explicit forms of Wd(x) and demonstrate that they can be expressed as combinations of d-1 hypergeometric functions of type d₋₁Fd₋₂ of argument x/dᵈ. These solutions are unique. We analyse them analytically and graphically. A combinatorially relevant, specific extension of Cd(n) for d even in the form Dd(n)=[∏ₚ ₌ ₀ᵈ⁻¹ p!/(n+p)!] [∏q ₌ ₀ᵈ/² ⁻ ¹ (2 n + 2 q)!/(2 q)!] is analyzed along the same lines.