On a q-extension of Mehta's eigenvectors of the finite Fourier transform   for q a root of unity
  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. 
Mesuma K. Atakishiyeva, Natig M. Atakishiyev and Tom H. Koornwinder