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Let ${\left(U_t\right)}_{t\ge 0}$ be a Brownian motion valued in the complex projective space $ℂP^{N-1}$. Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of $|U_t^1{|}^2$ and of $\left(\right|U_t^1{|}^2,\left|U_t^2{|}^2\right)$, and express them through Jacobi polynomials in the simplices of $ℝ$ and ${ℝ}^2$ respectively. More generally, the distribution of $\left(\right|U_t^1{|}^2,\cdots ,\left|U_t^k{|}^2\right),2\le k\le N-1$ may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group $𝒰(N-k+1)$ yet computations become tedious. We also revisit the approach initiated in [] and based on a...

Given an orthogonal projection P and a free unitary Brownian motion $Y={\left(Yₜ\right)}_{t\ge 0}$ in a W*-non commutative probability space such that Y and P are *-free in Voiculescu’s sense, we study the spectral distribution νₜ of Jₜ = PYₜPYₜ*P in the compressed space. To this end, we focus on the spectral distribution μₜ of the unitary operator SYₜSYₜ*, S = 2P - 1, whose moments are related to those of Jₜ via a binomial-type expansion already obtained by Demni et al. [Indiana Univ. Math. J. 61 (2012)]. In this connection,...

We shed some light on the inter-connections between different characterizations leading to the classical Meixner family. This allows us to give free analogs of both Sheffer's and Al-Salam and Chihara's characterizations in the classical case by the use of the free derivative operator. The paper closes with a discussion of the q-deformed case, |q| < 1.