The r-transform is a fundamental concept in noncommutative probability that provides a way to study and analyze the additive properties of noncommutative random variables. It acts as a generalization of the classical moment-generating functions by allowing us to connect the moments of a noncommutative random variable with its distribution through a functional calculus framework. This transformation is especially useful in the context of free probability theory, where it helps in understanding the behavior of sums of noncommutative random variables.
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