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

100%
Wojakowski Ł.
Banach Center Publications
|
2007
|
tom 78
|
nr 1
309-314
EN
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 $μ ⊞_T ν = T^{-1}(Tμ ⊞ Tν)$. 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 all s ≥ 1. This behaviour is similar to the free case, as in the original paper of Nica and Speicher [NS].
2
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Moments of measure orthogonalizing the 2-dimensional Chebyshev polynomials

100%
Wojakowski Ł.
Banach Center Publications
|
2006
|
tom 73
|
nr 1
429-433
EN
We calculate the moments $m_{k,0}$ of the measure orthogonalizing the 2-dimensional Chebyshev polynomials introduced by Koornwinder.
3
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Two-level t-deformation

100%
Wojakowski Ł.
Banach Center Publications
|
2010
|
tom 89
|
nr 1
313-322
EN
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.
4
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Remarks on Catalan and super-Catalan numbers

63%
Krystek A., Wojakowski Ł.
Banach Center Publications
|
2011
|
tom 96
|
nr 1
237-244
EN
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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