Generalized t-transformations of probability measures and deformed convolutions

In this paper, the generalized (two-parameterized) t-transformations on probability measures are introduced, in which the t-transformation of Bożejko and Wysoczański can be obtained as the special case, and the associated deformed convolutions are also investigated. We see that the generalized t-deformed free convolution can be realized as the conditionally free convolution of Bożejko, Leinert, and Speicher. We also study another special case of the generalized t-deformed free convolution, which is called the τ-free convolution, that gives an interpolation between the free and the Fermi convolutions.