*Article* **Some Properties Involving** *q***-Hermite Polynomials Arising from Differential Equations and Location of Their Zeros**

**Cheon-Seoung Ryoo <sup>1</sup> and Jungyoog Kang 2,\***


**Abstract:** Hermite polynomials are one of the Apell polynomials and various results were found by the researchers. Using Hermit polynomials combined with *q*-numbers, we derive different types of differential equations and study these equations. From these equations, we investigate some identities and properties of *q*-Hermite polynomials. We also find the position of the roots of these polynomials under certain conditions and their stacked structures. Furthermore, we locate the roots of various forms of *q*-Hermite polynomials according to the conditions of *q*-numbers, and look for values which have approximate roots that are real numbers.

**Keywords:** *q*-Hermite polynomials; zeros of *q*-Hermite polynomials; differential equation
