Appell polynomials and their relatives III. Conditionally free theory

TL;DR

This paper extends Appell polynomial theory to multivariate non-commutative c-free probability, establishing new polynomial families, their properties, and connections to operator algebra frameworks.

Contribution

It introduces and analyzes multivariate Appell-like polynomials in c-free probability, linking them to orthogonality, recursions, and Hilbert space models.

Findings

Defined c-free Appell polynomials as fixed points of a polynomial transformation.

Established recursions, generating functions, and martingale properties.

Connected c-free polynomials to Kailath-Segall polynomials and their combinatorics.

Abstract

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.