  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.
