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Topics on Meixner families

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Bożejko M., Demni N.

Banach Center Publications

2010

tom 89

nr 1

61-74

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.

Spectral distribution of the free Jacobi process associated with one projection

100%

Demni N., Hmidi T.

Colloquium Mathematicum

2014

tom 137

nr 2

271-296

Given an orthogonal projection P and a free unitary Brownian motion $Y = (Yₜ)_{t≥0}$ 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 use free stochastic calculus in order to derive a partial differential equation for the Herglotz transform μₜ. Then, we exhibit a flow ψ(t,·) valued in [-1,1] such that the composition of the Herglotz transform with the flow is governed by both the ones of the initial and the stationary distributions μ₀ and $μ_{∞}$. This enables us to compute the weights μₜ{1} and μₜ{-1} which together with the binomial-type expansion lead to νₜ{1} and νₜ{0}. Fatou's theorem for harmonic functions in the upper half-plane shows that the absolutely continuous part of νₜ is related to the nontangential extension of the Herglotz transform of μₜ to the unit circle. In the last part of the paper, we use combinatorics of noncrossing partitions in order to analyze the term corresponding to the exponential decay $e^{-nt}$ in the expansion of the nth moment of μₜ.

Ograniczanie wyników

1 Banach Center Publications

1 Colloquium Mathematicum

2 Demni N.

1 Bożejko M.

1 Hmidi T.

1 2014

1 2010

Wyniki wyszukiwania

Topics on Meixner families

Spectral distribution of the free Jacobi process associated with one projection