MULTIDIMENSIONAL CATALAN AND RELATED NUMBERS ASHAUSDORFF MOMENTS
Katarzyna Górska Karol A. Penson
Abstract: We study integral representation of the so-called -dimen- sional
Catalan numbers , defined by ,
, We prove that the ’s are the th Hausdorff power
moments of positive functions defined on . We construct exact and
explicit forms of and demonstrate that they can be expressed as combinations of
hypergeometric functions of type of argument . These solutions are
unique. We analyze them analytically and graphically. A combinatorially relevant, specific
extension of for even in the form
-dimen- sional
Catalan numbers 
, defined by !](files/33.2/HTML/33.2.8.abs2x.png)
,
, 
We prove that the 
’s are the 
th Hausdorff power
moments of positive functions 
defined on ![d
x ∈ [0,d ]](files/33.2/HTML/33.2.8.abs8x.png)
. We construct exact and
explicit forms of 
and demonstrate that they can be expressed as combinations of
hypergeometric functions of type 
of argument 
. These solutions are
unique. We analyze them analytically and graphically. A combinatorially relevant, specific
extension of 
for 
even in the form
![d∏-1 d∕2∏-1 (2n+2q)!
Dd(n) = [ (np+!p)!][ -(2q)!-]
p=0 q=0](files/33.2/HTML/33.2.8.abs15x.png)
-dimensional Catalan numbers, Hausdorff moment
problem.