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Decompositions of the free additive convolution

We introduce and study a new type of convolution of probability measures called the orthogonal convolution, which is related to the monotone convolution. Using this convolution, we derive alternating decompositions of the free additive convolution of compactly supported probability measures in free probability. These decompositions are directly related to alternating decompositions of the associated subordination functions. In particular, they allow us to compute free additive convolutions of compactly supported measures without using free cumulants or R-transforms. In simple cases, representations of the corresponding Cauchy transforms as continued fractions are obtained in a natural way. Moreover, this approach establishes a clear connection between convolutions and products associated with the main notions of independence (free, monotone and boolean) in noncommutative probability. Finally, our result leads to natural decompositions of the free product of rooted graphs.