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Abstract

Using the Nevanlinna representation of the reciprocal of the Cauchy transform of probability measures, we introduce a two-parameter transformation $U^{\mathbb T}$ of probability measures on the real line $\mathbb{R}$, 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 $\mathbb{T}$-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 $\mathbb{T}$-deformed free convolution is different from the convolution obtained as the deformed conditionally free convolution of Bożejko, Leinert and Speicher. Thus the $\mathbb{T}$ does not satisfy the Bożejko property.