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4 Banach Center Publications

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The $V_a$-deformation of the classical convolution

100%

Krystek A.

Banach Center Publications

2007

tom 78

nr 1

185-199

We study deformations of the classical convolution. For every invertible transformation T:μ ↦ Tμ, we are able to define a new associative convolution of measures by $μ{*_T}ν = T^{-1}(Tμ*Tν)$. We deal with the $V_a$-deformation of the classical convolution. We prove the analogue of the classical Lévy-Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the $V_a$-deformed classical convolution and give the orthogonal polynomials associated to the limiting measure.

On some generalization of the t-transformation

100%

Krystek A.

Banach Center Publications

2010

tom 89

nr 1

165-187

Using the Nevanlinna representation of the reciprocal of the Cauchy transform of probability measures, we introduce a two-parameter transformation $U^{𝕋}$ of probability measures on the real line ℝ, which is another possible generalization of the t-transformation. Using that deformation we define a new convolution by deformation of the free convolution. The central limit measure with respect to the 𝕋-deformed free convolutions is still a Kesten measure, but the Poisson limit depends on the two parameters and is different from the Poisson measures for (a,b)-deformation. We also show that the 𝕋-deformed free convolution is different from the convolution obtained as the deformed conditionally free convolution of Bożejko, Leinert and Speicher. Thus the 𝕋 does not satisfy the Bożejko property.

Remarks on the boolean convolution and Kerov's α-transformation

100%

Krystek A.

Banach Center Publications

2006

tom 73

nr 1

285-298

This paper consists of two parts. The first part is devoted to the study of continuous diagrams and their connections with the boolean convolution. In the second part we investigate the rectangular Young diagrams and respective discrete measures. We recall the definition of Kerov's α-transformation of diagrams, define the α-transformation of finitely supported discrete measures and generalize the notion of the α-transformation.

Remarks on Catalan and super-Catalan numbers

63%

Krystek A. , Wojakowski Ł.

Banach Center Publications

2011

tom 96

nr 1

237-244

In this article we discuss the Catalan and super-Catalan (or Schröder) numbers. We start with some combinatorial interpretations of those numbers. We study two probability measures in the context of free probability, one whose moments are super-Catalan, and another, whose even moments are super-Catalan and odd moments are zero. With the use of the latter we also show some new formulae for evaluation of the Catalans in terms of super-Catalans and vice-versa.