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Mehta has shown that eigenvectors of the N × N finite Fourier transform can be written in terms of the standard Hermite eigenfunctions of the quantum harmonic oscillator (1987 J. Math. Phys. 28 781). Here, we construct a one-parameter family of q-extensions of these eigenvectors, based on the continuous q-Hermite polynomials of Rogers. In the limit when q - 1 these q-extensions coincide with Mehta's eigenvectors, and in the continuum limit as N - [?] they give rise to q-extensions of eigenfunctions of the Fourier integral transform. [ABSTRACT FROM AUTHOR]

We define a two-dimensional (2D) Fourier transform that self-reproduces a one-parameter family of bivariate Hermite functions; these are eigenfunctions of a Hamiltonian differential operator of second order, whose exponential is that transform. We find explicit forms of the bivariate raising and lowering partial differential operators of first degree for the eigenfunctions of this 2D Fourier transform. [ABSTRACT FROM AUTHOR]

It is shown that the continuous q-Hermite polynomials for q, a root of unity, have simple transformation properties with respect to the classical Fourier transform. This result is then used to construct q-extended eigenvectors of the finite Fourier transform in terms of these polynomials. [ABSTRACT FROM AUTHOR]

We prove that the classical Fourier-transform operator \widehat{\cal F} intertwines two q-difference equations for the continuous q-Hermite polynomials Hn(x|q) of Rogers, which are associated with the two distinct sets of values for the parameter q: 0 < q < 1 and 1 < q < [?]. [ABSTRACT FROM AUTHOR]

We construct a new model of the quantum oscillator, whose energy spectrum is equally-spaced and lower-bound, whereas the spectra of position and of momentum are a denumerable non-degenerate set of points in [[?] 1, 1] that depends on the deformation parameter q [?] (0, 1). We provide its explicit wavefunctions, both in position and momentum representations, in terms of the discrete q-Hermite polynomials. We build a Hilbert space with a unique measure, where an analogue of the fractional Fourier transform is defined in order to govern the time evolution of this discrete oscillator. In the limit when q - 1[?], one recovers the ordinary quantum harmonic oscillator. [ABSTRACT FROM AUTHOR]