Quantum algebra from generalized q-Hermite polynomials

TL;DR

This paper explores the properties of generalized q-Hermite polynomials, establishing their orthogonality, integral representations, and their connection to quantum algebra structures, specifically $ ext{su}_q(1,1)$, through q-oscillator operators.

Contribution

It introduces new continuous orthogonality relations, q-integral representations, and a realization of quantum algebra using generalized q-Hermite polynomials and q-oscillators.

Findings

Established orthogonality relations for generalized q-Hermite polynomials

Derived q-integral representations and Poisson kernel evaluations

Realized quantum algebra $ ext{su}_q(1,1)$ via q-deformed oscillators

Abstract

In this paper, we discuss new results related to the generalized discrete $q$-Hermite II polynomials $ \tilde h_{n,\alpha}(x;q)$, introduced by Mezlini et al. in 2014. Our aim is to give a continuous orthogonality relation, a $q$-integral representation and an evaluation at unity of the Poisson kernel, for these polynomials. Furthermore, we introduce $q$-Schr\"{o}dinger operators and give the raising and lowering operator algebra corresponding to these polynomials. Our results generate a new explicit realization of the quantum algebra $\mathsf{su}_{q}(1, 1)$, using the generators associated with a $q$-deformed generalized para-Bose oscillator.