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While dealing with the constant-strength magnetic Laplacian on the annulus,we complete J. Peetre's work. In particular, the eigenspaces associated withits discrete spectrum are true-polyanalytic spaces with respect to theinvariant Cauchy-Riemann operator, and we write down explicit formulas fortheir reproducing kernels. The latter are expressed by means of the fourthJacobi theta function and of its logarithmic derivatives when the magneticfield strength is an integer. Under this quantization condition, we also derivethe transformation rule satisfied by the reproducing kernel under theautomorphism group of the annulus.

Polyanalytic Reproducing Kernels on the Quantized Annulus