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9 by Si Si. 5. White noise functionals of higher chaos degree (−3) (−3) To fix the idea, we shall discuss dualities related to H3 . Let ϕ be in H3 . Its kernel function F (u1 , u2 , u3 ) is found in the Sobolev space K −2 (R3 ). The S-transform U (ξ) = (Sϕ)(ξ) can, therefore, be expressed in the form U (ξ) = F (u1 , u2 , u3 )ξ(u1 )ξ(u2 )ξ(u3 )du3 . Our method with the idea that le passage du fini ` a l’infini leads us to consider the class of normal functionals, namely we are interested in the following forms of degree three.

Barhoumi, H. Ouerdiane & A. 1. If we choose β = 1 and c2 = α ∈]1, 2[, then the measure να := ν in partition of unity satisfied by the Gamma, Meixner and Pascal coherent vectors {Ωz }z∈C is given by να (dz) = 1 Iα−1 (2|z|)Kα−1 2|z| dz. π 5. Complex Hypercontractivity for the Meixner class The quantum mechanical harmonic oscillator is essentially the Weyl representation of the Lie algebra associated to the Euclidean motion group. In Fock-Bargmann model, it can be described by the quadruple10 {HL2 (C, ̺), ∂, ∂ † , H} where HL2 (C, ̺) = 1/2 f : C → C, holomorphic, ∂f (z) = ∂ f (z), ∂z f 2 := C ∂ † f (z) = zf (z), |f (z)|2 ̺(dz) Hf (z) = z <∞ , ∂ f (z), ∂z and ̺(dz) = π1 e−zz dz is the complex one-dimensional Gaussian measure.

2010, to appear. jp We formulate a class of differential equations for white noise operators including the quantum white noise derivatives and obtain a general form of the solutions. As application we characterize intertwining operators appearing in the implementation problem for the canonical commutation relation. Keywords: Fock space, implementation problem, quantum white noise, quantum white noise derivative, Wick product 1. ” This aspect has been widely accepted explicitly or implicitly [1,2,7,8,16,17], however, the systematic study has not actively developed mostly because of the cumbersome property of the annihilation and creation operators being formulated as (unbounded) operator-valued distributions.

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