Mathematics – Operator Algebras
Scientific paper
2008-03-29
Illinois J. Math. 53 (2009), 39-66
Mathematics
Operator Algebras
A major revision: same theorems, different emphasis
Scientific paper
We extend to the multivariate non-commutative context the descriptions of a "once-stripped" probability measure in terms of Jacobi parameters, orthogonal polynomials, and the moment generating function. The corresponding map Phi on states was introduced previously by Belinschi and Nica. We then relate these constructions to the c-free probability theory, which is a version of free probability for algebras with two states, introduced by Bozejko, Leinert, and Speicher. This theory includes as two extreme cases the free and Boolean probability theories. The main objects in the paper are the analogs of the Appell polynomial families in the two state context. They arise as fixed points of the transformation which takes a polynomial family to the associated polynomial family (in several variables), and their orthogonality is also related to the map Phi above. In addition, we prove recursions, generating functions, and factorization and martingale properties for these polynomials, and describe the c-free version of the Kailath-Segall polynomials, their combinatorics, and Hilbert space representations.
No associations
LandOfFree
Appell polynomials and their relatives III. Conditionally free theory does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Appell polynomials and their relatives III. Conditionally free theory, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Appell polynomials and their relatives III. Conditionally free theory will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-511629