A Hörmander–Fock space

In a recent paper, we used a basic decomposition property of polyanalytic functions of order 2 in one complex variable to characterize solutions of the classical (Formula presented.) -problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander–Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by (Formula presented.) leading to a special entire function (Formula presented.) that is used to express the kernel function of the Hörmander–Fock space. We present also an example of a special function belonging to the class Mittag-Leffler (ML) introduced recently by Alpay et al. and apply a Bochner–Minlos type theorem to this function, thus motivating further connections with the theory of stochastic processes.