On q-extended eigenvectors of the integral and finite Fourier transforms

Mehta has shown that eigenvectors of the N x 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