Quantum algebra from generalized q-Hermite polynomials

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.