Appell Polynomials and Their Relatives Iii. Conditionally Free Theory

ABSTRACT. This paper describes the analogs of the Appell polynomial families in the context of algebras with two states, also called the c-free probability theory, introduced by Bożejko, Speicher, and Leinert. This theory includes as two extreme cases the free and Boolean probability theories. We prove recursions, generating functions, and factorization and martingale properties for these polynomials. We characterize the orthogonal c-free Appell polynomials in terms of the map introduced previously by Belinschi and Nica. We also note that the free Appell polynomials are exactly the fixed points of the transformation which takes polynomials of the first kind to polynomials of the second kind, and generalize these notions to higher dimensions. Finally, we describe the c-free version of the Kailath-Segall polynomials, their combinatorics, and Hilbert space representations.