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

Abstract

In this work, we investigate the sequence of monic q-Hermite I-Sobolev type orthogonal polynomials of higher-order, denoted by $\{\mathbb {H}_{n}(x;q)\}_{n\ge 0}$, which are orthogonal with respect to the following non-standard inner product involving q-differences:

$$\begin{aligned} \langle p,q\rangle _{\lambda }=\int _{-1}^{1}f\left( x\right) g\left( x\right) (qx,-qx;q)_{\infty }d_{q}(x)+\lambda \,(\mathscr {D}_{q}^{j}f)(\alpha )(\mathscr {D}_{q}^{j}g)(\alpha ), \end{aligned}$$

where $\lambda$ belongs to the set of positive real numbers, $\mathscr {D}_{q}^{j}$ denotes the j-th q -discrete analogue of the derivative operator, $q^j\alpha \in \mathbb {R}\backslash (-1,1)$, and $(qx,-qx;q)_{\infty }d_{q}(x)$ denotes the orthogonality weight with its points of increase in a geometric progression. Connection formulas between these polynomials and standard q-Hermite I polynomials are deduced. The basic hypergeometric representation of $\mathbb {H}_{n}(x;q)$ is obtained. Moreover, for certain real values of $\alpha$ satisfying the condition $q^j\alpha \in \mathbb {R}\backslash (-1,1)$, we present results concerning the location of the zeros of $\mathbb {H}_{n}(x;q)$ and perform a comprehensive analysis of their asymptotic behavior as the parameter $\lambda$ tends to infinity.