A curious $q$-analogue of Hermite polynomials

TL;DR

This paper introduces a new family of $q$-Hermite polynomials, explores their unique properties, and reveals connections with $q$-Fibonacci and $q$-Lucas polynomials, extending classical combinatorial formulas.

Contribution

The paper presents a novel family of $q$-Hermite polynomials and establishes their relationships with $q$-Fibonacci and $q$-Lucas polynomials, generalizing known combinatorial identities.

Findings

New family of $q$-Hermite polynomials introduced

Established connection with $q$-Fibonacci and $q$-Lucas polynomials

Generalized Touchard-Riordan formula

Abstract

Two well-known $q$-Hermite polynomials are the continuous and discrete $q$-Hermite polynomials. In this paper we consider a new family of $q$-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with $q$-Fibonacci and $q$-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.