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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.

A family of sequences of binomial type

Młotkowski W., Romanowicz A.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

401--408

For a delta operator aD – bDp+1 we find the corresponding polynomial sequence of binomial type and relations with Fuss numbers. In the case of D – 1/2D2 we show that the corresponding Bessel-Carlitz polynomials are moments of the convolution semigroup of inverse Gaussian distributions.We also find probability distributions vt, t > 0, for which {yn(t)}, the Bessel polynomials at t, is the moment sequence.

Limit measures related to the conditionally free convolution

Hinz M., Młotkowski W.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 1

75-81

We describe the limit measures for some class of deformations of the free convolution, introduced by A. D. Krystek and Ł. J. Wojakowski. In particular, we provide a counterexample to a conjecture from their paper.