We study integral representation of the so-called d-dimensional Catalan numbers Cd(n), defined by [Πd−1p=0 p!=(n + p)!] (dn)!, 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 analyze tchem analytically and graphically. A combinatorially relevant, specific extension of Cd(n) for d even in the form Dd(n) = [wzór] is analyzed along the same lines.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.