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Two-level t-deformation

Łukasz Jan Wojakowski — 2010

Banach Center Publications

In the present paper we define and study the properties of a deformation of measures and convolutions that works in a similar way to the U t deformation of Bożejko and Wysoczański, but in its definition operates on two levels of Jacobi coefficients of a measure, rather than on one.

The Lévy-Khintchine formula and Nica-Speicher property for deformations of the free convolution

Łukasz Jan Wojakowski — 2007

Banach Center Publications

We study deformations of the free convolution arising via invertible transformations of probability measures on the real line T:μ ↦ Tμ. We define new associative convolutions of measures by μ T ν = T - 1 ( T μ T ν ) . We discuss infinite divisibility with respect to these convolutions, and we establish a Lévy-Khintchine formula. We conclude the paper by proving that for any such deformation of free probability all probability measures μ have the Nica-Speicher property, that is, one can find their convolution power μ T s for...

Remarks on Catalan and super-Catalan numbers

Anna Dorota KrystekŁukasz Jan Wojakowski — 2011

Banach Center Publications

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.

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