Fixed points of the Berezin transform on Fock-type spaces

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