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.

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
$\mu {\u229e}_{T}\nu ={T}^{-1}\left(T\mu \u229eT\nu \right)$.
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 ${\mu}^{{\u229e}_{T}s}$ for...

We calculate the moments ${m}_{k,0}$ of the measure orthogonalizing the 2-dimensional Chebyshev polynomials introduced by Koornwinder.

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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