  We study fluctuations of linear statistics in Polyanalytic Ginibre ensembles,
a family of point processes describing planar free fermions in a uniform
magnetic field at higher Landau levels. Our main result is asymptotic normality
of fluctuations, extending a result of Rider and Vir\'ag. As in the analytic
case, the variance is composed of independent terms from the bulk and the
boundary. Our methods rely on a structural formula for polyanalytic polynomial
Bergman kernels which separates out the different pure $q$-analytic kernels
corresponding to different Landau levels. The fluctuations with respect to
these pure $q$-analytic Ginibre ensembles are also studied, and a central limit
theorem is proved. The results suggest a stabilizing effect on the variance
when the different Landau levels are combined together.
