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2 Probability and Mathematical Statistics

2 2013

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The probability measure corresponding to 2-plane trees

Młotkowski W., Penson K. A.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

255--264

We study the probability measure μ0 for which the moment sequence is (3n/n) 1/n+1 . We prove that μ0 is absolutely continuous, find the density function and prove that μ0 is infinitely divisible with respect to the additive free convolution.

Multidimensional Catalan and related numbers as Hausdorff moments

Górska K., Penson K. A.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

265--274

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.