#### Abstract

The objective of this study is to find -differential equations of higher order related to -modified derangements’ polynomials and confirm the structure of approximation roots. Furthermore, it states several symmetric properties of -differential equations of higher order and the special properties of the approximation roots of -modified derangements’ polynomials.

#### 1. Introduction

##### 1.1. Bernoulli Differential Equation

is an equation, where is any real number, and and are continuous functions on the interval. Amongst all of the differential equations, the Bernoulli differential equation converts nonlinear equations into linear equations. If or , the above equation is linear, and if not, the equation is nonlinear. By substituting , the Bernoulli differential equation can be reduced to a linear differential equation. Then, it concludes to a linear equation for . This equation can be applied to problems related to nonlinear differential equations, equations about the population expressed in logistic equations, or Verhulst equations, physics, etc.

If in (1), the Bernoulli differential equation,has a solution which is the generating function of modified derangements’ polynomials, see [1, 2].

For , the modified derangements’ numbers and polynomials can be expressed asrespectively.

Table 1 is the first few examples of the modified derangements’ numbers and polynomials .

Based on the above concept and -numbers, we consider the -Bernoulli differential equation of the first order . In addition, it is possible to assume that the -modified derangements’ polynomials are a solution of the following -differential equation of the first order when in (1).

The aim of this study is to find the -Bernoulli differential equation with the solution of -modified derangements’ polynomials. Also, we derive characteristic properties by visualizing the approximation roots of -modified derangements’ polynomials.

Several mathematicians discovered -differential equations by using special polynomials as a solution and studying their properties and identities, see [3–7]. The -differential equation based on -Hermit polynomials were studied by Hermoso, Huertas, and Lastra in [8]. In [9, 10], phenomena of roots for various kinds of polynomials related to differential equations were researched by Ryoo. Furthermore, various properties of polynomials using -series, -derivative, -distribution, and so on were found, see [9, 11–13].

To lay out the foundation for achieving the goal of this study, the following summarizes the definitions and theorems and makes arrangements.

Jackson introduced the -number, which plays an important role in -calculus, see [4, 14]. Based on the discovery of -number, useful results are studied in -series, -special functions, quantum algebras, -discrete distribution, -differential equations, -calculus, etc., see [15, 16]. Here, we briefly review several concepts of -calculus which we need for this study.

Let with . The numberis called -number, see [6, 16]. We note that . In particular, for , is called -integer.

The -Gaussian binomial coefficients are defined bywhere and are nonnegative integers, see [17]. For , the value is 1 since the numerator and the denominator are both empty products. One notes and .

*Definition 1. *Let be any complex number with . The -exponential functions can be expressed as follows:(i) (ii)For , where is the -Pochhammer symbol.

We note that , see [12, 16].

Theorem 1. *From Definition 1, we note that*

*Definition 2. *The -derivative of a function with respect to is defined byand .

We can prove that is differentiable at zero, and it is clear that , see [6, 14–16]. From Definition 2, there are some formulae for -derivative.

Theorem 2. *From Definition 2, we note that*

Our ultimate purpose is to find the solution of -modified derangements’ polynomials by observing various -differential equations of higher order. In Section 2, we define the -modified derangements’ numbers and polynomials, mention several forms of -differential equations of higher order, and check its associated symmetric properties. Lastly, in Section 3, by observing the values of -modified derangements’ numbers, the approximation roots of -modified derangements’ polynomials will be shown, and several conjectures for those numbers and polynomials will be organized.

#### 2. Various Types of -Differential Equations of Higher Order Associated with the -Modified Derangements’ Polynomials

In this section, -modified derangements’ polynomials are defined, and various kinds of -differential equation of higher order associated with these polynomials are introduced. Moreover, we find several symmetric properties of -differential equation of higher order.

*Definition 3. *For and , the -modified derangements’ polynomials is defined with the following generating function as follows:From Definition 3, we note thatHere, we define as the -modified derangements’ numbers. The -modified derangements’ numbers has the relation with the polynomials of -derangements, see [1].

Theorem 3. *The -modified derangements’ polynomials is a solution of the -differential equation which can be given as*

*Proof. *When , the generating function of the -modified derangements’ polynomials haveThe left-hand side of (12) can be changed toUsing (12) and (13), applying the Cauchy product rule and comparing the coefficients of both sides of result equation, we haveApplying -derivative in , we find a relation between and asSubstituting of (15) to the left-hand side of (14), we achieve the shown result.

Corollary 1. *When in Theorem 3, one holds thatwhere is the modified derangements’ polynomials.*

Theorem 4. *The solution of the following -differential equation of higher order is the -modified derangements’ polynomials which can be given as follows:*

*Proof. *Using a property of -exponential function in the generating function of the -modified derangements’ polynomials, we findSuppose that in (18). Then, we haveFrom the power series of -exponential functions, the left-hand side of (19) can be transformed asComparing (19) and (20), we haveApplying -derivative in , we obtain a relation such asFrom (21) and (22), we find Combining (21) and (23), we complete the desired result.

Corollary 2. *Let in Theorem 4. Then, we obtainwhere is the modified derangements’ polynomials.*

Theorem 5. *The -modified derangements’ polynomials is a solution of the following -differential equation of higher order which is combined with the -modified derangements’ numbers:*

*Proof. *Initially, we apply the -derivative after substituting instead of in the generating function of the -modified derangements’ polynomials . Then, we obtainFrom (26), we haveFrom the generating function of the -modified derangements’ polynomials , we also haveComparing (27) and (28), we obtainApplying relation (22) in (29), we deriveand this equation gives us the required result.

Corollary 3. *Setting in Theorem 5, we obtainwhere is the modified derangements’ number and is the modified derangements’ polynomials.*

Theorem 6. *The -modified derangements’ polynomials is a solution of the -differential equation of higher order which iswhere is the -modified derangements’ numbers.*

*Proof. *apply the -derivative after substituting instead of in . Then, we haveIn a similar way to find (27) and (28), we obtainFrom (34) and (35), we haveHere, we find a relation between and as follows:Using (37) in the left-hand side of (36), we haveBy (38), it is possible to finish the proof of Theorem 6.

Corollary 4. *Putting in Theorem 6, one holds thatwhere is the modified derangements’ numbers and is the modified derangements’ polynomials.*

From now on, we take several suitable forms to find symmetric properties of -differential equation of higher order. To obtain various symmetric properties of -differential equation of higher order which combines other numbers and polynomials, the below forms are the basic forms.

Theorem 7. *Let , , and . Then, we obtain*

*Proof. *To obtain a symmetric property of -differential equation of higher order for -modified derangements’ polynomials , we consider a form asFrom the form , we haveFrom (42) and (43), we findReplacing (37) to (44), we obtainFrom (45), we complete the proof of Theorem 7.

Corollary 5. *Setting in Theorem 7, we have*

Corollary 6. *consider in Theorem 7; then, we get*

Theorem 8. *Let , , and . Then, we have*

*Proof. *Suppose form asUsing , the desired result can be obtained in a similar way to the proof of Theorem 7. Therefore, we omit the proof process.

Corollary 7. *Setting in Theorem 8, we have*

Corollary 8. *consider in Theorem 8; then, we get*

#### 3. Visualization of the Approximation Roots for -Modified Derangements’ Polynomials

This section handles the approximation roots of the -modified derangements’ polynomial. To confirm several conjectures for -modified derangements, we show the structure of the approximation roots of those polynomials. We use MATHEMATICA in order to obtain the pictures and the results of calculation.

Based on the generating function of -modified derangements’ numbers, are found as follows:

From the above -modified derangements’ numbers, Table 2 shows the approximation values of which appear with changes in values of . In Table 1, as the value of increases, we can observe that the approximation value of -modified derangements’ numbers also increases.

By using Table 2, we can see the location of shown by varying and , as shown in Figure 1. The nonnegative integers of the *x*-axis represent the values of in Figure 1. Here, the lines display variations of the approximation values for -modified derangements’ numbers. The blue dots, the yellow squares, and the green rhombuses in Figure 1 are the approximation values of -modified derangements’ numbers when , respectively. For example, the blue dot above the value 6 on the real axis shows the approximation value of , and the yellow square indicates the approximation value of .

From Table 2 and Figure 1, we can find the following conjecture.

Conjecture 1. *As increases and approaches 1, the approximation value of increases.*

Using the generating function of -modified derangements’ polynomials, are found as follows:

Based on the above polynomial, we find special properties of approximations values that appear depending on the value of of the -modified derangements’ polynomials. Here, we set the values of to 0.01, 0.001, and 0.0001, respectively, because the extremely small value of represents the properties of the -modified derangements’ polynomials. We experiment with the -modified derangements’ polynomials for two purposes. The first is to check the structure of the approximation roots and find the polygon that is most similar to it. The second is to find the approximation values of the real number among the approximation roots appearing in the -modified derangements’ polynomials.

To obtain the first purpose, Figure 2 shows the structure of the approximate roots of -modified derangements’ polynomials. Here, the range of is from 0 to 50, and the value of changes. In Figure 2, the value of in (a) is 0.01, the value of in (b) is 0.001, and the value of in (c) is 0.0001. A characteristic that can be confirmed in the structure of the approximation roots in Figure 2(c) is that, as increases, one approximation root close to the origin continues to accumulate. The structures of the approximation roots shown in Figure 2 seem to become more circular as increases.

Based on the results of Figure 2, we find Figure 3. Figure 3 shows the structure of approximation roots when the value of is 50. In Figure 3, the value of in (a) is 0.01, the value of in (b) is 0.001, and the value of in (c) is 0.0001. Here, we set the red dots to indicate the approximation roots, the blue dots to indicate the center of the circle, and the blue lines to connect the red dots. Also, to obtain the blue lines closest to the approximation roots, the real roots are excluded.

Table 3 is the result of calculating the exact values in Figure 3. From Table 3, we can see that, as the value of gets smaller, the structure of the approximation roots has a shape closer to a circle, and the radius of the circle gets closer to 1.

From Figures 2 and 3 and Table 3, we make the following conjecture.

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

Conjecture 2. *As becomes smaller than 0.001, the structure of the approximation roots of located on a circle will have a radius close to 1.*

From now on, based on Figure 2, we find approximation roots by dividing them into real and imaginary roots. Since our concern is the real roots, it can be obtained as shown in Table 4 by finding and expressing the real roots.

We can find a special property for -modified derangements’ polynomials in Table 4. Table 4 shows that only odd-order polynomials of have approximation real roots even if the value of changes, and only one approximation real root exists. Therefore, we can make the following assumptions.

Conjecture 3. *-modified derangements’ polynomials have only one approximation real root in odd-order polynomials regardless of the change in the -number.*

#### 4. Conclusion

We organized the -differential equation with the solution of -modified derangements’ polynomials and find the corresponding symmetric properties. We also looked at the distribution of approximation roots of -modified derangements’ polynomials and their phenomena. As a result, we can make some assumptions and think that the related research should be continued.

#### Data Availability

The data used to support the findings of this study can be obtained from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.