On a q-extension of Mehta's eigenvectors of the finite Fourier transform for q a root of unity

TL;DR

This paper explores how continuous q-Hermite polynomials at roots of unity transform under Fourier analysis, enabling the construction of q-extended eigenvectors of the finite Fourier transform.

Contribution

It introduces a novel method to construct q-extended eigenvectors of the finite Fourier transform using properties of q-Hermite polynomials at roots of unity.

Findings

q-Hermite polynomials have simple Fourier transform properties at roots of unity

Constructed explicit q-extended eigenvectors of the finite Fourier transform

Enhanced understanding of q-extensions in Fourier analysis

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.