## Abstract

The key result in the paper concerns two transformations, Φ : (*ρ*, *ψ*) ↦ *φ* and 𝔹_{t} : *ψ* ↦ *φ*, where *ρ*, *ψ*, *φ* are states on the algebra of non-commutative polynomials, or equivalently joint distributions of *d*-tuples of non-commuting operators. These transformations are related to free probability: if ⊞ is the free convolution operation, and {*ρ _{t}*} is a free convolution semigroup, we show that

The maps {𝔹_{t}} were introduced by Belinschi and Nica as a semigroup of transformations such that 𝔹_{1} is the bijection between infinitely divisible distributions in Boolean and free probability theories. They showed that for *γ _{t}* the free heat semigroup and Φ the Boolean version of the Kolmogorov representation for infinitely divisible measures,

The more general map Φ[*ρ*, *ψ*] comes, not from free probability, but from the theory of two-state algebras, also called the conditionally free probability theory, introduced by Bożejko, Leinert, and Speicher. Orthogonality of the c-free versions of the Appell polynomials, investigated in [Anshelevich, J. Math.], is closely related to the map Φ. On the other hand, one can describe explicitly the action of all the transformations above on free Meixner families, which provides clues to their general behavior. Besides the evolution equation, other results include the positivity of the map Φ[*ρ*, *ψ*] and descriptions of its fixed points and range.

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