ON q-EXTENSIONS OF MEHTA'S EIGENVECTORS OF THE FINITE FOURIER TRANSFORM

Abstract

Mehta has shown that eigenvectors [Formula: see text] of the finite Fourier transform with the matrix [Formula: see text], 0 ≤ j, k ≤ N-1, can be defined in terms of the classical Hermite functions [Formula: see text] as [Formula: see text], where [Formula: see text]. We argue that the finite Fourier transform [Formula: see text] does actually govern also some q-extensions of Mehta's eigenvectors [Formula: see text], associated with certain well-known orthogonal q-polynomial families. For the pairs of the continuous q-Hermite and q-1-Hermite polynomials, the Rogers–Szegő and Stieltjes–Wigert polynomials, and the discrete q-Hermite polynomials of types I and II such links are explicitly derived. In the limit when the base q → 1 these q-extensions coincide with Mehta's eigenvectors [Formula: see text], whereas in the continuous limit (i.e. when the parameter N → ∞) they correspond to the classical Fourier integral transforms between the above-mentioned pairs of q-polynomial families.