**4. Distribution and Pattern of Zeros of** *q***-Hermite Polynomials**

In this section, we examine the distribution and pattern of zeros of *q*-Hermite polynomials **H***n*,*q*(*x*) according to the change in degree *n*. Based on these results, we present a problem that needs to be approached theoretically. Many mathematicians now explore concepts more easily than in the past by using software. These experiments allow them to quickly create and visualize new ideas, review properties of various figures, as well as find and guess patterns. This numerical survey is particularly interesting since it helps them understand the basic concepts and solve numerous problems. Here, we use MATHEMATICA to find Figures 2–4 and approximate roots for *q*-Hermite polynomials.

The *q*-Hermite polynomials **H***n*,*q*(*x*) can be explicitly determined; see [21,22]. First, several examples are given, as follows.

$$\begin{aligned} \mathbf{H}\_{0,q}(\mathbf{x}) &= 1, \\ \mathbf{H}\_{1,q}(\mathbf{x}) &= -\frac{2}{-1+q} + \frac{2q^x}{-1+q}, \\ \mathbf{H}\_{2,q}(\mathbf{x}) &= -2 + \frac{4}{(-1+q)^2} - \frac{8q^x}{(-1+q)^2} + \frac{4q^{2x}}{(-1+q)^2}, \\ \mathbf{H}\_{3,q}(\mathbf{x}) &= \frac{4}{(-1+q)^3} - \frac{24q}{(-1+q)^3} + \frac{12q^2}{(-1+q)^3} + \frac{12q^x}{(-1+q)^3} - \frac{24q^{2x}}{(-1+q)^3} \\ &+ \frac{8q^{3x}}{(-1+q)^3} + \frac{24q^{1+x}}{(-1+q)^3} - \frac{12q^{2+x}}{(-1+q)^3}. \end{aligned} \tag{52}$$

We observe the distribution of zeros of the *q*-Hermite polynomials **H***n*,*q*(*x*) = 0. In Figure 2, plots for the zeros of the *<sup>q</sup>*-Hermite polynomials **<sup>H</sup>***n*,*q*(*x*) for *<sup>n</sup>* <sup>=</sup> 20 and *<sup>x</sup>* <sup>∈</sup> <sup>R</sup> are as follows.

**Figure 2.** Zeros of **H***n*,*q*(*x*).

In the top-left picture of Figure 2, we choose *n* = 20 and *q* = 3/10. In the top-right picture of Figure 2, we consider conditions which are *n* = 20 and *q* = 5/10. We can find the bottom-left picture of Figure 2, when we consider *n* = 20 and *q* = 7/10. If we consider *n* = 20 and *q* = 9/10, then we can observe the bottom-right picture of Figure 2.

Stacks of zeros of the *q*-Hermite polynomials, **H***n*,*q*(*x*), for 1 ≤ *n* ≤ 20 from a 3-D structure are presented as Figure 3.

**Figure 3.** Stacks of zeros of **H***n*,*q*(*x*), 1 ≤ *n* ≤ 20.

It is the left picture of Figure 3, when we consider *q* = 1/2. Additionally, if we consider *q* = 9/10, we can obtains the right picture of Figure 3.

Our numerical results for the approximate solutions of real zeros of the *q*-Hermite polynomials, **<sup>H</sup>***n*,*q*(*x*), with *<sup>q</sup>* <sup>=</sup> 1/2 and *<sup>x</sup>* <sup>∈</sup> <sup>R</sup> are displayed in Tables <sup>1</sup> and 2.


**Table 1.** Numbers of real and complex zeros of **H***n*, <sup>1</sup> 2 (*x*) .

The plot structures of real zeros of the *q*-Hermite polynomials, **H***n*,*q*(*x*), for 1 ≤ *n* ≤ 20 are presented in Figure 4.

**Figure 4.** Stacks of zeros of **H***n*,*q*(*x*), 1 ≤ *n* ≤ 20.

In the left picture of Figure 4, we choose *q* = 5/10. For *q* = 9/10, the right side of Figure 4 is presented. Next, we calculated an approximate solution that satisfies *Hn*,*q*(*x*) = 0, *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>. The results are shown in Table 2.


**Table 2.** Approximate solutions of **<sup>H</sup>***n*,*q*(*x*) = 0, *<sup>x</sup>* <sup>∈</sup> <sup>R</sup> .

## **5. Conclusions and Discussion**

In this paper, we derive a few solutions of special forms containing *q*-Hermit polynomials and find several properties of differential equations for these polynomials. Moreover, we find approximate values of real zeros for *q*-Hermit polynomials and analyze the structure of roots for these polynomials in a special condition from 3D.

We also identified the structure of *q*-Hermit polynomials under special several conditions. These conditions change the structure of the roots and the form of polynomials, and further research needs to be done on finding various properties. In addition, by simulating the structure of roots for Hermit polynomials through various methods using the results of this paper and multiple software, it is also thought that the characteristics of the roots' structure for higher-order equations will evolve into one area.

**Author Contributions:** Conceptualization, C.-S.R.; methodology, J.K.; writing—original draft preparation, C.-S.R.; writing—review and editing, J.K.; funding acquisition, J.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and Future Planning(No. 2017R1E1A1A03070483).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare that they have no conflicts of interest to report regarding the present study.
