Berezin transform on the harmonic Fock space

We show that the Berezin transform associated to the harmonic Fock (Segal–Bargmann) space on Cn has an asymptotic expansion analogously as in the holomorphic case. The proof involves a computation of the reproducing kernel, which turns out to be given by one of Horn's hypergeometric functions of two variables, and an ad hoc determination of the asymptotic behaviour of the resulting integrals, to which the ordinary stationary phase method is not directly applicable.