  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 $\overline{\partial}$-problem for given analytic and polyanalytic
data. Our approach suggested the study of a special reproducing kernel Hilbert
space that we call the H\"ormander-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 $\eta=(\eta_n)_{n\geq 0}$ leading to a special
entire function $\mathsf{E}(z)$ that is used to express the kernel function of
the H\"ormander-Fock space. We present also an example of a special function
belonging to the class 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.
