Some Classes of Generating Functions for the Laguerre and Hermite Polynomials

Cohen, M. E.

doi:10.2307/2006433

Cited by 1 publication

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“…Proof. The series in the LHS of (4.6) is an instance of equation 1.2 from [3] where it is claimed without any detail that it converges in D. This claim can be checked as follows: the derivation rule ( [17])…”

Section: An Integral Representationmentioning

confidence: 99%

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Free Jacobi Process Associated with One Projection: Local Inverse of the Flow

Демни

2015

Complex Anal. Oper. Theory

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Abstract. We pursue the study started in [8] of the dynamics of the spectral distribution of the free Jacobi process associated with one orthogonal projection. More precisely, we use Lagrange inversion formula in order to compute the Taylor coefficients of the local inverse around z = 0 of the flow determined in [8]. When the rank of the projection equals 1/2, the obtained sequence reduces to the moment sequence of the free unitary Brownian motion. For general ranks in (0, 1), we derive a contour integral representation for the first derivative of the Taylor series which is a major step toward the analytic extension of the flow in the open unit disc. Reminder and motivationThe free Jacobi process (J t ) t≥0 was introduced in [6] as the large-size limit of the Hermitian matrix Jacobi process ([9]). It is built as the radial part of the compression of the free unitary Brownian motion (Y t ) t≥0 ([1]) by two orthogonal projections {P, Q}:In this definition, both families of operators {P, Q} and (Y t ) t≥0 are * -free (in Voiculescu's sense) in a von Neumann algebra A endowed with a finite trace τ and a unit 1. When J t is considered as a positive operator valued in the compressed algebra (P A P, τ /τ (P )), its spectral distribution µ t 1 is a probability distribution supported in [0, 1] and the positive real number τ (P )µ t {1} encodes the general position property for {P, Y t QY * t } ([4], [13]). On the other hand, the couple of papers [7] and [8] aim to determine the Lebesgue decomposition of µ t when both projections coincide P = Q. In particular, when τ (P ) = 1/2, a complete description was given in [7] (see Corollary 3.3 there and [13] for another proof): at any time t ≥ 0, µ t coincides with the spectral distribution of Y 2t + Y * 2t + 21 4 considered as a positive operator in (A , τ ) (the spectral distribution of Y 2t , say η 2t , was described in [2], Proposition 10). For an arbitrary rank τ (P ) ∈ (0, 1), only the discrete part in the Lebesgue decomposition of µ t was determined in [8] (see Theorem 1.1). As to its absolutely-continuous part with respect to Lebesgue measure in [0,1], it was related to that of the spectral distribution, say ν t , of the unitary operatorActually, the density of the former distribution is related to the density of the latter through the Caratheodory extension of the Riemann map of the cut planeThe key ingredient leading to this partial description is a flow ψ t so far defined and exploited in an interval of the form (−1, z t ) for some z t ∈ (0, 1), t > 0. When τ (P ) = 1/2, ψ t is a one-to-one map from a Jordan domain onto the open unit disc D and its compositional inverse coincides, up to a Cayley transform, with the Herglotz transform of η 2t . For arbitrary ranks, ψ t is locally invertible and further information on ν t (which in turn provide information on µ t ) necessitates the investigation of the analytic extension of of the local inverse of ψ t in D. Moreover, it was shown in [8] that ν t converges weakly to a probability measure ν ∞ whose support disconnects as ...

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Section: An Integral Representationmentioning

confidence: 99%

“…Coming into the derivation of the RHS of (4.6), we specialize equation (1.2) in [3] to b = 0, v = m + 1, x = 2(m + 1)t, a = 1/(m + 1) in order to get:…”

Section: An Integral Representationmentioning

confidence: 99%

Free Jacobi Process Associated with One Projection: Local Inverse of the Flow

Демни

2015

Complex Anal. Oper. Theory

View full text Add to dashboard Cite

show abstract

Some Classes of Generating Functions for the Laguerre and Hermite Polynomials

Cited by 1 publication

References 8 publications

Free Jacobi Process Associated with One Projection: Local Inverse of the Flow

Free Jacobi Process Associated with One Projection: Local Inverse of the Flow

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