  The main content of this work is devoted to study various explicit family of
special functions generalizing the famous 2D Sleipain functions, founded in
1960's by D. Slepian and his co-authors. As a consequence, many desirable
spectral properties of the corresponding weighted finite Fourier transform are
deduced from the rich literature. In particular, similar aspect related to
Slepian's seminal papers is the investigation of differential operators that
commute with appropriate integral operators are given. Finally, we provided the
reader with some analytic expressions for the Fourier transforms of the Disk
polynomials and the two variables Gegenbauer polynomials.
