q-Extension of Mehta's eigenvectors of the finite Fourier transform forq, a root of unity

Abstract

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.

Exact solution of Chern-Simons-matter matrix models with characteristic/orthogonal polynomials

“…Notice that we will need the polynomials for q roots of unity. While they are better understood for q < 1 (or q > 1), they also hold for q root of unity [23]. Notice for example, that the q-binomial coefficient is a polynomial in q and hence also well-defined [23].…”

“…While they are better understood for q < 1 (or q > 1), they also hold for q root of unity [23]. Notice for example, that the q-binomial coefficient is a polynomial in q and hence also well-defined [23]. In general, the whole four-parameter Askey-Wilson polynomials are studied also for q root of unity [24] (the Stieltjes-Wigert are at the bottom of the Askey classification scheme).…”