MULTIDIMENSIONAL CATALAN AND RELATED NUMBERS AS HAUSDORFF MOMENTS

作者：Gorska Katarzyna^*; Penson Karol A

来源：Probability and Mathematical Statistics, 2013, 33(2): 265-274.

• d-dimensional Catalan numbers
• Hausdorff moment problem

We study integral representation of the so-called d-dimensional Catalan numbers C-d(n), defined by [Pi(d-1)(p=0)p!/(n + p)!](dn)!, d = 2, 3, ..., n = 0, 1, ... We prove that the C-d(n)'s are the nth Hausdorff power moments of positive functions W-d(x) defined on x is an element of [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 F-d-1(d-2) of argument x/d(d). These solutions are unique. We analyze them analytically and graphically. A combinatorially relevant, specific extension of C-d(n) for d even in the form
D-d(n) = [Pi(d-1)(p=0) p!/(n+p)!][Pi(d/2-1)(q=0) (2n+2q)!/(2q)!]
is analyzed along the same lines.

• 出版日期2013