Complex Hermite functions as Fourier-Wigner transform

D-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization

“…The normalized vacuum state ϕ 0,0 , satisfying a i ϕ 0,0 = 0, i = 1, 2, is simply the constant function h 0,0 (z) = 1. These polynomials have been discussed extensively in the literature (see, for example, [6,19,25,34], and very recently in [4,29,30,31,32,33]). Their expression can be directly inferred from (3.2)…”

“…The first one involves a family of biorthogonal polynomials, named deformed complex Hermite polynomials, various properties of which have been worked out during the past years (see for instance [8,16] and references therein). Their construction involves a deformation of the well-known bivariate complex Hermite polynomials [4,25,29,30,31,32,33] using finite-dimensional irreducible representations of the group GL(2, C) of invertible 2 × 2 matrices with complex entries and reveals interesting aspects of these representations. The second example introduces families of vectors and operators in the underlying Hilbert space built in the same same way as standard coherent states, as orbits in the Hilbert space of the projective Weyl-Heisenberg group.…”