On zero behavior of higher-order Sobolev-type discrete q-Hermite I orthogonal polynomials

In this work, we investigate the sequence of monic q-Hermite I-Sobolev type orthogonal polynomials of higher-order, denoted as $\{{\mathbb{H}}_n(x;q){\}}_{n\ge 0}$, which are orthogonal with respect to the following non-standard inner product involving q-differences: \begin{equation*} \langle p,q\rangle_{λ}=\int_{-1}^{1}f\left( x\right) g\left(x\right) (qx,-qx;q){\infty }d{q}(x)+λ,(\mathscr{D}{q}^{j}f)(α)(\mathscr{D}{q}^{j}g)(α), \end{equation*} where $\alpha \in \mathbb{R}\mathrm{\backslash }(-1,1)$, $\lambda$ belongs to the set of positive real numbers, ${\mathcal{D}}_q^j$ denotes the $j$-th $q$-discrete analogue of the derivative operator, and $(qx,-qx;q{)}_{\mathrm{\infty }}{d}_q(x)$ denotes the orthogonality weight with its points of increase in a geometric progression. We proceed to obtain the hypergeometric representation of ${\mathbb{H}}_n(x;q)$ and explicit expressions for the corresponding ladder operators. From the latter, we obtain a novel kind of three-term recurrence formula with rational coefficients associated with these polynomial family. Moreover, for certain real values of $\alpha$, we present some results concerning the location of the zeros of ${\mathbb{H}}_n(x;q)$ and we perform a comprehensive analysis of their asymptotic behavior as the parameter $\lambda$ varies from zero to infinity.