On Fourier integral transforms for $q$-Fibonacci and $q$-Lucas polynomials

TL;DR

This paper investigates two families of $q$-Fibonacci and $q$-Lucas polynomials, revealing their simple transformation properties under classical Fourier integral transforms, and explores their relation to $q$-extensions of Hermite polynomials.

Contribution

The paper provides a detailed analysis of two new $q$-polynomial families and their behavior under Fourier transforms, connecting them to classical and $q$-Hermite polynomials.

Findings

Both $q$-Fibonacci and $q$-Lucas polynomials exhibit simple Fourier transform properties.

These polynomials are related to $q$-extensions of Hermite polynomials.

The study offers new insights into the structure of $q$-polynomials and their transforms.

Abstract

We study in detail two families of $q$-Fibonacci polynomials and $q$-Lucas polynomials, which are defined by non-conventional three-term recurrences. They were recently introduced by Cigler and have been then employed by Cigler and Zeng to construct novel $q$-extensions of classical Hermite polynomials. We show that both of these $q$-polynomial families exhibit simple transformation properties with respect to the classical Fourier integral transform.