Fixed points of the Berezin transform on Fock-type spaces

We study the fixed points of the Berezin transform on the Fock-type spaces F m 2 F_m^2 with the weight e − | z | m , m > 0 e^{-|z|^m}, m > 0 . It is known that the Berezin transform is well-defined on the polynomials in z z and z ¯ \overline {z} . In this paper we focus on the polynomial fixed points and we show that these polynomials must be harmonic, except possibly for countably many m ∈ ( 0 , ∞ ) m \in (0, \infty ) . We also show that, in some particular cases, the fixed point polynomials are harmonic for all m m .