Multilinear function series and transforms in free probability theory

The algebra Mul〚B〛 of formal multilinear function series over an algebra B and its quotient SymMul〚B〛 are introduced, as well as corresponding operations of formal composition. In the setting of Mul〚B〛, the unsymmetrized R- and T-transforms of random variables in B-valued noncommutative probability spaces are introduced. These satisfy properties analogous to the usual R- and T-transforms (the latter being just the reciprocal of the S-transform), but describe all moments of a random variable, not only the symmetric moments. The partially ordered set of noncrossing linked partitions is introduced and is used to prove properties of the unsymmetrized T-transform.