  We consider and provide an accurate study for the fractional Zernike
functions on the punctured unit disc, generalizing the classical Zernike
polynomials and their associated $\beta$-restricted Zernike functions. Mainly,
we give the spectral realization of the latter ones and show that they are
orthogonal $L^2$-eigenfunctions for certain perturbed magnetic (hyperbolic)
Laplacian. The algebraic and analytic properties for the fractional Zernike
functions to be established include the connection to special functions, their
zeros, their orthogonality property, as well as the differential equations,
recurrence, and operational formulas they satisfy. Integral representations are
also obtained. Their regularity as poly-meromorphic functions is discussed and
their generating functions including a bilinear one of "Hardy--Hille type" are
derived. Moreover, we prove that a truncated subclass defines a complete
orthogonal system in the underlying Hilbert space giving rise to a specific
Hilbertian orthogonal decomposition in terms of a second class of generalized
Bergman spaces.
