A new characterization of $q$-Chebyshev polynomials of the second kind

In this work, we introduce the notion of $\mathcal{U}_{(q,\mu)}$-classical orthogonal polynomials, where $\mathcal{U}_{(q,\mu)}$ is the degree raising shift operator defined by $\mathcal{U}_{(q,\mu)}:=x(xH_q+q^{-1}I_{\mathcal{P}})+\mu H_q,$ where $\mu$ is a nonzero free parameter, $I_{\mathcal{P}}$ represents the identity operator on the space of polynomials $\mathcal{P}$, and $H_q$ is the $q$-derivative one. We show that the scaled $q$-Chebychev polynomials of the second kind ${\hat{U}}_{n}(x, q), n\geq0$, are the only $\mathcal{U}_{(q,\mu)}$-classical orthogonal polynomials.