We describe the class of probability measures whose moments are given in terms of the Aval numbers. They are expressed as the multiplicative free convolution of measures corresponding to the ballot numbers $(m-k)/(m+k) \binom{m+k}{m}$.
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We present an operator-valued version of the conditionally free product of states and measures, which in the scalar case was studied by Bożejko, Leinert and Speicher. The related combinatorics and limit theorems are provided.
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We study the nonnegative product linearization property for polynomials with eventually constant Jacobi parameters. For some special cases a necessary and sufficient condition for this property is provided.
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We compute moments of the measures $(ϖ^{⊠ p})^{⊞ t}$, where ϖ denotes the free Poisson law, and ⊞ and ⊠ are the additive and multiplicative free convolutions. These moments are expressed in terms of the Fuss-Narayana numbers.
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We describe the limit measures for some class of deformations of the free convolution, introduced by A. D. Krystek and Ł. J. Wojakowski. In particular, we provide a counterexample to a conjecture from their paper.
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We study relations between the Boolean convolution and the symmetrization and the pushforward of order 2. In particular we prove that if μ₁,μ₂ are probability measures on [0,∞) then $(μ₁ ⨄ μ₂)^{s} = μ₁^{s} ⨄ μ₂^{s}$ and if ν₁,ν₂ are symmetric then $(ν₁ ⨄ ν₂)^{(2)} = ν₁^{(2)} ⨄ ν₂^{(2)}$. Finally we investigate necessary and sufficient conditions under which the latter equality holds.
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We compute the moments and free cumulants of the measure $ρ_t: = π_t ⊠ π_t$, where $π_t$ denotes the free Poisson law with parameter t > 0. We also compute free cumulants of the symmetrization of $ρ_t$. Finally, we introduce the free symmetrization of a probability measure on ℝ and provide some examples.
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