Berezin transforms attached to landau levels on the complex projective space pⁿ(c)

We construct coherent states for each eigenspace of a magnetic Lapla-cian on the complex projective n-space in order to apply a quantization-dequantization method. Doing so allows to define the Berezin transform for these spaces. We then establish a formula for this transform as a function of the Fubini–Study Laplacian in a closed form involving of a terminating Kampé de Fériet function. For the lowest spherical Landau level on the Riemann sphere the obtained formula reduces to the one derived by Berezin himself.