Polynomials in $c$-free random variables with applications to free denoising

We study distributions of polynomials in conditionally free (c-free) random variables, a notion of independence for two-state noncommutative probability spaces introduced by Bozejko, Leinert and Speicher. To this end we establish recursive relations between the joint Boolean cumulants of c-free random variables, analogous to previously found recursions for Boolean cumulants of free random variables. The algebraic reformulation of these recursions on the free associative algebra provides an effective formal machinery for the computation of the moment generating functions and thus the distributions of arbitrary self-adjoint polynomials in c-free random variables. As an application of a recent observation, our approach can be used to determine conditional expectations of the form $E[a|P(a,b)]$, where $P(a,b)$ is a self-adjoint polynomial in free (in the sense of Voiculescu) random variables $a,b$. We illustrate this with an example where $P(a,b)=i[a,b]$. Finally we define orthogonal projections that formally play the role of conditional expectations in the framework of c-freeness and share some properties with the conditional expectations of free variables. In particular they can be used to re-derive by purely algebraic methods the formula of Popa and Wang for the $\Sigma$-transform for the c-free multiplicative convolution.