We employ the recently established basis (the two-variable Hermite–Gaussian function) of the generalized Bargmann space (BGBS) [Phys. Lett. A 303, 311 (2002) ] to study the generalized form of the fractional Fourier transform (FRFT). By using the technique of integration within an ordered product of operators and the bipartite entangled-state representations, we derive the generalized generating function of the BGBS with which the undecomposable kernel of the two-dimensional FRFT [also named complex fractional Fourier transform (CFRFT)] is obtained. This approach naturally shows that the BGBS is just the eigenfunction of the CFRFT.
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