On a New Characterization of the True-Poly-Analytic Bargmann Spaces

We consider a novel bounded integral transform with a kernel function being the n-th polyanalytic Intissar–Hermite polynomial. We provide a concrete description of its range, shown to be a reproducing kernel Hilbert space for which we provide an explicit closed formula of its reproducing kernel. The limit of these ranges leads, in a precise sense, to the so-called van Eijndhoven–Meyers Bargmann space, which also corresponds to the limit case. The null space of the considered transform is also characterized. It gives rise in particular to new integral representation of the true-polyanalytic Bargmann space by means of the polyanalytic Intissar–Hermite integral transform.