**Proof.**

(i) We consider form *A* as follows:

$$A := \frac{t^2 \varepsilon\_{p,q}(t\mathbf{x}) \varepsilon\_{p,q}(tX) \text{COS}\_{p,q}(ty) \text{COS}\_{p,q}(tY)}{\left(\varepsilon\_{p,q}(at) - 1\right) \left(\varepsilon\_{p,q}(bt) - 1\right)}\tag{66}$$

From form *A*, we find

$$\begin{split} A &= \frac{t}{\varepsilon\_{p,q}(at) - 1} c\_{p,q}(tx) \text{CO}\_{p,q}(ty) \frac{t}{\varepsilon\_{p,q}(bt) - 1} c\_{p,q}(tX) \text{CO}\_{p,q}(tY) \\ &= \sum\_{n=0}^{\infty} a^{n-1} \\_ ^ {\infty} B\_{n,p,q} \left( \frac{x}{a}, \frac{y}{a} \right) \frac{t^n}{[n]\_{p,q}!} \sum\_{n=0}^{\infty} b^{n-1} \\_ ^ {\infty} B\_{n,p,q} \left( \frac{X}{b}, \frac{Y}{b} \right) \frac{t^n}{[n]\_{p,q}!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} a^{n-k-1} b^{k-1} \\_ ^ {\infty} B\_{n-k,p,q} \left( \frac{x}{a}, \frac{y}{a} \right) \\_ ^ {\infty} B\_{k,p,q} \left( \frac{X}{b}, \frac{Y}{b} \right) \right) \frac{t^n}{[n]\_{p,q}!} . \end{split} \tag{67}$$

and form *A* of Equation (66) can be transformed into the following:

$$\begin{split} A &= \frac{t}{\varepsilon\_{p,q}(bt) - 1} \varepsilon\_{p,q}(tx) \text{COS}\_{p,q}(ty) \frac{t}{\varepsilon\_{p,q}(at) - 1} \varepsilon\_{p,q}(tX) \text{COS}\_{p,q}(tY) \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{n-k-1} a^{k-1} \text{\textdegree B}\_{n-k,p,q} \left( \frac{\chi}{b}, \frac{y}{b} \right) \text{\textdegree B}\_{k,p,q} \left( \frac{X}{a}, \frac{Y}{a} \right) \right) \frac{t^{n}}{[n]\_{p,q}!} . \end{split} \tag{68}$$

Using the comparison of coefficients in Equations (67) and (68), we find the desired result. (ii) If we assume form *B* as follows:

$$B := \frac{t^2 \varepsilon\_{p,q}(t\mathbf{x}) \varepsilon\_{p,q}(t\mathbf{X}) SIN\_{p,q}(t\mathbf{y}) SIN\_{p,q}(t\mathbf{y})}{\left(\varepsilon\_{p,q}(at) - 1\right) \left(\varepsilon\_{p,q}(bt) - 1\right)},\tag{69}$$

then, we find Theorem 9 (*ii*) in the same manner.

**Corollary 11.** *Setting a* = 1 *in Theorem 9, the following holds:*

$$\begin{split} (i) \quad & \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{k-1} \subset B\_{n-k,p,q} \left( \mathbf{x}, \mathbf{y} \right) \subset B\_{k,p,q} \left( \frac{\mathbf{X}}{b}, \frac{\mathbf{Y}}{b} \right) \\ & \quad = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{n-k-1} \subset B\_{n-k,p,q} \left( \frac{\mathbf{x}}{b}, \frac{\mathbf{y}}{b} \right) \subset B\_{k,p,q} \left( \mathbf{X}, \mathbf{Y} \right), \\ (ii) \quad & \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{k-1} \subset B\_{n-k,p,q} \left( \mathbf{x}, \mathbf{y} \right) \operatorname{\mathbf{S}}\_{k,p,q} \left( \frac{\mathbf{X}}{b}, \frac{\mathbf{Y}}{b} \right) \\ & \quad = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{n-k-1} \operatorname{\mathbf{S}} B\_{n-k,p,q} \left( \frac{\mathbf{x}}{b}, \frac{\mathbf{y}}{b} \right) \operatorname{\mathbf{S}} B\_{k,p,q} \left( \mathbf{X}, \mathbf{Y} \right). \end{split} \tag{70}$$

**Corollary 12.** *If p* = 1 *in Theorem 9, then we have*

$$\begin{split} (i) \quad & \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_q a^{n-k-1} b^{k-1} \circ B\_{n-k,q} \left( \frac{\mathbf{x}}{a}, \frac{\mathbf{y}}{a} \right) \circ B\_{k,q} \left( \frac{\mathbf{X}}{b}, \frac{\mathbf{Y}}{b} \right) \\ & = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_q b^{n-k-1} a^{k-1} \circ B\_{n-k,q} \left( \frac{\mathbf{x}}{b}, \frac{\mathbf{y}}{b} \right) \circ B\_{k,q} \left( \frac{\mathbf{X}}{a}, \frac{\mathbf{Y}}{a} \right), \\ (ii) \quad & \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_q a^{n-k-1} b^{k-1} \circ B\_{n-k,q} \left( \frac{\mathbf{x}}{a}, \frac{\mathbf{y}}{a} \right) \circ B\_{k,q} \left( \frac{\mathbf{X}}{b}, \frac{\mathbf{Y}}{b} \right) \\ & = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_q b^{n-k-1} a^{k-1} \circ B\_{n-k,q} \left( \frac{\mathbf{x}}{b}, \frac{\mathbf{y}}{b} \right) \circ B\_{k,q} \left( \frac{\mathbf{X}}{a}, \frac{\mathbf{Y}}{a} \right), \end{split} \tag{71}$$

*where <sup>C</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *denotes the q-cosine Bernoulli polynomials and <sup>S</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *denotes the q-sine Bernoulli polynomials.*

**Corollary 13.** *Putting p* = 1 *and q* → 1*, one holds:*

$$\begin{split} (i) \quad & \sum\_{k=0}^{n} \binom{n}{k} a^{n-k-1} b^{k-1} \circ B\_{n-k} \left( \frac{x}{a}, \frac{y}{a} \right) \circ B\_k \left( \frac{X}{b}, \frac{Y}{b} \right) \\ & = \sum\_{k=0}^{n} \binom{n}{k} b^{n-k-1} a^{k-1} \circ B\_{n-k} \left( \frac{x}{b}, \frac{y}{b} \right) \circ B\_k \left( \frac{X}{a}, \frac{Y}{a} \right), \\ (ii) \quad & \sum\_{k=0}^{n} \binom{n}{k} a^{n-k-1} b^{k-1} \circ B\_{n-k} \left( \frac{x}{a}, \frac{y}{a} \right) \circ B\_k \left( \frac{X}{b}, \frac{Y}{b} \right) \\ & = \sum\_{k=0}^{n} \binom{n}{k} b^{n-k-1} a^{k-1} \circ B\_{n-k} \left( \frac{x}{b}, \frac{y}{b} \right) \circ B\_k \left( \frac{X}{a}, \frac{Y}{a} \right), \end{split} \tag{72}$$

*where <sup>C</sup>Bn*(*<sup>x</sup>*, *y*) *is the cosine Bernoulli polynomials and <sup>S</sup>Bn*(*<sup>x</sup>*, *y*) *is the sine Bernoulli polynomials.*

Theorem 9 is a basic symmetric property of (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials. We aim to find several symmetric properties by mixing (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials.

**Theorem 10.** *For nonzero integers a and b, we have*

$$\begin{split} &\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} a^{n-k-1} b^{k-1} \subset \mathcal{B}\_{n-k,p,q} \left( \frac{\mathcal{X}}{a}, \frac{\mathcal{Y}}{a} \right) \,\_{\mathbb{S}} \mathcal{B}\_{k,p,q} \left( \frac{\mathcal{X}}{b}, \frac{\mathcal{Y}}{b} \right) \\ &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{n-k-1} a^{k-1} \subset \mathcal{B}\_{n-k,p,q} \left( \frac{\mathcal{X}}{b}, \frac{\mathcal{Y}}{b} \right) \,\_{\mathbb{S}} \mathcal{B}\_{k,p,q} \left( \frac{\mathcal{X}}{a}, \frac{\mathcal{Y}}{a} \right) . \end{split} \tag{73}$$

**Proof.** We assume form *C* by mixing the (*p*, *q*)-cosine function with the (*p*, *q*)-sine function, such as the following:

$$\mathbb{C} := \frac{t^2 \mathfrak{e}\_{p,q}(t\mathbf{x}) \mathfrak{e}\_{p,q}(tX) \mathbb{C} \mathrm{OS}\_{p,q}(ty) \mathrm{SIN}\_{p,q}(tY)}{\left(\mathfrak{e}\_{p,q}(at) - 1\right) \left(\mathfrak{e}\_{p,q}(bt) - 1\right)}. \tag{74}$$

Form *C* of the above equation can be changed into

$$\begin{split} \mathcal{C} &= \frac{t}{\varepsilon\_{p,q}(at) - 1} c\_{p,q}(tx) \text{COS}\_{p,q}(ty) \frac{t}{\varepsilon\_{p,q}(bt) - 1} c\_{p,q}(tX) SIN\_{p,q}(tY) \\ &= \sum\_{n=0}^{\infty} a^{n-1} \,\_{\subset} B\_{n,p,q} \left( \frac{\mathfrak{x}}{a}, \frac{\mathfrak{y}}{a} \right) \frac{t^n}{[n]\_{p,q}!} \sum\_{n=0}^{\infty} b^{n-1} \,\_{\subset} B\_{n,p,q} \left( \frac{\mathfrak{X}}{b}, \frac{\mathfrak{Y}}{b} \right) \frac{t^n}{[n]\_{p,q}!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} a^{n-k-1} b^{k-1} \,\_{\subset} B\_{n-k,p,q} \left( \frac{\mathfrak{x}}{a}, \frac{\mathfrak{Y}}{a} \right) \,\_{\subset} B\_{k,p,q} \left( \frac{\mathfrak{X}}{b}, \frac{\mathfrak{Y}}{b} \right) \right) \frac{t^n}{[n]\_{p,q}!} . \end{split} \tag{75}$$

or, equivalently:

$$\begin{split} \mathbb{C} &= \frac{t}{\varepsilon\_{p,q}(bt) - 1} e\_{p,q}(t\mathbf{x}) \mathbb{C} \mathbf{O}\_{p,q}(ty) \frac{t}{\varepsilon\_{p,q}(at) - 1} e\_{p,q}(t\mathbf{X}) S \, lN\_{p,q}(t\mathbf{y}) \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{n-k-1} a^{k-1} c \, B\_{n-k,p,q} \left( \frac{\mathbf{x}}{b^{\prime}} \frac{\mathbf{y}}{b^{\prime}} \right) \,\_{\mathbb{S}} B\_{k,p,q} \left( \frac{\mathbf{X}}{a^{\prime}} \frac{\mathbf{y}}{a} \right) \right) \frac{t^{n}}{[n]\_{p,q}!} . \end{split} \tag{76}$$

By comparing transformed Equations (75) and (76), we determine the result of Theorem 10.

**Corollary 14.** *If a* = 1 *in Theorem 10, then we find*

$$\begin{split} &\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{k-1} \subset {}^{B}B\_{n-k,p,q} \left( \mathbf{x}, \mathbf{y} \right) \, {}\_{S}B\_{k,p,q} \left( \frac{\mathbf{X}}{\mathbf{b}}, \frac{\mathbf{Y}}{\mathbf{b}} \right) \\ &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{n-k-1} \subset {}^{B}B\_{n-k,p,q} \left( \frac{\mathbf{x}}{\mathbf{b}}, \frac{\mathbf{y}}{\mathbf{b}} \right) \, {}\_{S}B\_{k,p,q} \left( \mathbf{X}, \mathbf{y} \right) \, {}\_{S}B\_{n-k,p,q} \left( \frac{\mathbf{y}}{\mathbf{b}}, \mathbf{y} \right) . \end{split} \tag{77}$$

**Corollary 15.** *Setting p* = 1 *in Theorem 10, one holds:*

$$\begin{split} &\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_q a^{n-k-1} b^{k-1} \mathbb{C} B\_{n-k,q} \left( \frac{\underline{X}}{a}, \frac{\underline{Y}}{a} \right) \,\_S B\_{k,q} \left( \frac{\underline{X}}{b}, \frac{\underline{Y}}{b} \right) \\ &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_q b^{n-k-1} a^{k-1} \mathbb{C} B\_{n-k,q} \left( \frac{\underline{X}}{b}, \frac{\underline{Y}}{b} \right) \,\_S B\_{k,q} \left( \frac{\underline{X}}{a}, \frac{\underline{Y}}{a} \right) \,\_S \end{split} \tag{78}$$

*where <sup>C</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *is the q-cosine Bernoulli polynomials and <sup>S</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *is the q-sine Bernoulli polynomials.*

**Corollary 16.** *Assigning p* = 1 *and q* → 1 *in Theorem 10, the following holds:*

$$\begin{split} &\sum\_{k=0}^{n} \binom{n}{k} a^{n-k-1} b^{k-1} \subset \mathcal{B}\_{n-k} \left( \frac{\chi}{a}, \frac{\underline{y}}{a} \right) \, \_S B\_k \left( \frac{X}{b}, \frac{Y}{b} \right) \\ &= \sum\_{k=0}^{n} \binom{n}{k} b^{n-k-1} a^{k-1} \subset \mathcal{B}\_{n-k} \left( \frac{\chi}{b}, \frac{\underline{y}}{b} \right) \, \_S B\_k \left( \frac{X}{a}, \frac{Y}{a} \right) \, \_V \end{split} \tag{79}$$

*where <sup>C</sup>Bn*(*<sup>x</sup>*, *y*) *is the cosine Bernoulli polynomials and <sup>S</sup>Bn*(*<sup>x</sup>*, *y*) *is the sine Bernoulli polynomials.* *Symmetry* **2020**, *12*, 885

**Theorem 11.** *Let a and b be nonzero integers. Then, we derive*

$$\begin{split} &\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} a^{n-k-1} b^{k-1} \subset & B\_{n-k,p,q} \left( b x, \frac{y}{a} \right) \,\_S B\_{k,p,q} \left( a X, \frac{Y}{b} \right) \\ &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{n-k-1} a^{k-1} \subset & B\_{n-k,p,q} \left( a x, \frac{y}{b} \right) \,\_S B\_{k,p,q} \left( b X, \frac{Y}{a} \right) \,\_S \end{split} \tag{80}$$

**Proof.** Let us consider form *D* containing *a* and *b* in the (*p*, *q*)-exponential functions as

$$D := \frac{t^2 e\_{p,q}(abtx) e\_{p,q}(abtX) \text{COS}\_{p,q}(ty) SIN\_{p,q}(tY)}{\left(e\_{p,q}(at) - 1\right) \left(e\_{p,q}(bt) - 1\right)}. \tag{81}$$

From the above form *D* , we can obtain

$$\begin{split} D &= \frac{t}{\varepsilon\_{p,q}(at) - 1} \varepsilon\_{p,q}(abtx) \mathbf{C}OS\_{p,q}(ty) \frac{t}{\varepsilon\_{p,q}(bt) - 1} \varepsilon\_{p,q}(abtX)SIN\_{p,q}(tY) \\ &= \sum\_{n=0}^{\infty} a^{n-1} \\_ \_ \mathbf{C}B\_{n,p,q}\left(bx, \frac{y}{a}\right) \frac{t^n}{[n]\_{p,q}!} \sum\_{n=0}^{\infty} b^{n-1} \\_ \_ \mathbf{S}B\_{n,p,q}\left(aX, \frac{Y}{b}\right) \frac{t^n}{[n]\_{p,q}!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_ \_ \_ \right. \left. a^{n-k-1} b^{k-1} \\_ \mathbf{C}B\_{n-k,p,q}\left(bx, \frac{y}{a}\right) \, \_ \_ \mathbf{S}B\_{k,p,q}\left(aX, \frac{Y}{b}\right) \right) \frac{t^n}{[n]\_{p,q}!} \end{split} \tag{82}$$

and

$$\begin{split} D &= \frac{t}{\varepsilon\_{p,q}(bt) - 1} \varepsilon\_{p,q}(abtx) \text{COS}\_{p,q}(ty) \frac{t}{\varepsilon\_{p,q}(at) - 1} \varepsilon\_{p,q}(abtx) SIN\_{p,q}(tY) \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{n-k-1} a^{k-1} \text{c}B\_{n-k,p,q} \left( a\mathbf{x}, \frac{y}{b} \right) \text{s}B\_{k,p,q} \left( b\mathbf{X}, \frac{Y}{a} \right) \right) \frac{t^n}{[n]\_{p,q}!} . \end{split} \tag{83}$$

By observing Equations (82) and (83) which are made by form *D*, we prove Theorem 11.

**Corollary 17.** *Setting a* = 1 *in Theorem 11, the following holds:*

$$\begin{split} &\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{k-1} \ll\_{\mathbb{C}} B\_{n-k,p,q} \left( b \mathbf{x}\_{\prime} \mathbf{y} \right) \,\_{\mathbb{S}} B\_{k,p,q} \left( \mathbf{X}\_{\prime} \frac{\mathbf{y}}{\mathbf{b}} \right) \\ &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} b^{n-k-1} \ll B\_{n-k,p,q} \left( \mathbf{x}\_{\prime} \frac{\mathbf{y}}{\mathbf{b}} \right) \,\_{\mathbb{S}} B\_{k,p,q} \left( b \mathbf{X}\_{\prime} \mathbf{y} \right) \,\_{\mathbb{S}} \end{split} \tag{84}$$

**Corollary 18.** *If p* = 1 *in Theorem 11, then we obtain*

*n* ∑ *k*=0 %*nk*&*qan*−*k*−1*bk*−1*CBn*−*k*,*<sup>q</sup> bx*, *ya <sup>S</sup>Bk*,*<sup>q</sup> aX*, *Yb* = *n* ∑ *k*=0 %*nk*&*qbn*−*k*−1*ak*−1*CBn*−*k*,*<sup>q</sup> ax*, *yb <sup>S</sup>Bk*,*<sup>q</sup> bX*, *Ya* , (85)

*where <sup>C</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *is the q-cosine Bernoulli polynomials and <sup>S</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *is the q-sine Bernoulli polynomials.* **Corollary 19.** *Let p* = 1 *and q* → 1 *in Theorem 11. Then one holds*

$$\begin{split} &\sum\_{k=0}^{n} \binom{n}{k} a^{n-k-1} b^{k-1} \, \_\subset B\_{n-k} \left( b \ge \frac{\underline{y}}{a} \right) \, \_S B\_k \left( a \ge \frac{\underline{y}}{b} \right) \\ &= \sum\_{k=0}^{n} \binom{n}{k} b^{n-k-1} a^{k-1} \, \_\subset B\_{n-k} \left( a \ge \frac{\underline{y}}{b} \right) \, \_\subset B\_k \left( b \ge \frac{\underline{y}}{a} \right) \, \_\subset \end{split} \tag{86}$$

*where <sup>C</sup>Bn*(*<sup>x</sup>*, *y*) *is the cosine Bernoulli polynomials and <sup>S</sup>Bn*(*<sup>x</sup>*, *y*) *is the sine Bernoulli polynomials.*

Next, we investigate the structure of approximate roots in (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials. Based on the theorems above, (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials have symmetric properties. Thus, we assume that the approximate roots of (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials also have symmetric properties as well. We aim to identify the stacking structure of the roots from the specific (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials found in Section 3.

First, the structure of approximate roots in the (*p*, *q*)-cosine Bernoulli polynomials is illustrated in Figure 1 when *y* = 5, *q* = 0.9, and the value of *p* changes. Figure 1 reveals the pattern of the roots in the (*p*, *q*)-cosine Bernoulli polynomials when *p* = 0.5. In addition, the approximate roots appear when *n* changes from 1 to 30. The red points become closer together when *n* is 30 and *n* becomes smaller as illustrated by the blue points. Based on the graphs with real and imaginary axes, the (*p*, *q*)-cosine Bernoulli polynomials are symmetric.

**Figure 1.** Stacking structure of approximate roots in the (*p*, *q*)-cosine Bernoulli polynomials when *p* = 0.5, *q* = 0.9, and *y* = 5.

Here, we aim to confirm that changes in the value of the (*p*, *q*)-cosine Bernoulli polynomials changes the structure of the approximate roots as the value changes. The structure of the approximate roots in polynomials when *p* = 1 and *q* changes, can be found in the *q*-cosine Bernoulli polynomials (see [11]).

Figure 2 below illustrates the stacking structure of the approximate roots of the (*p*, *q*)-cosine Bernoulli polynomials fixed at *p* = 0.1, *q* = 0.5 and *y* = 5 when 1 ≤ *n* ≤ 30. Compared with Figure 1, Figure 2 displays a wider distribution of the approximate roots. The range of the left picture in Figure 1 is −15 < Re *x* < 15 and the range of the left picture in Figure 2 is −50 < Re *x* < 50. The structure of the approximate roots of *p* = 0.1 when *n* = 30 is wider on the real axis compared to when *p* = 0.5. The right-hand graphs in Figures 1 and 2 also reveal the same distribution. In addition, as *n* increases, the structure of the approximate roots appears symmetric.

**Figure 2.** Stacking structure of approximate roots in the (*p*, *q*)-cosine Bernoulli polynomials when *p* = 0.1, *q* = 0.9, and *y* = 5.

Next, we examine the stacking structure of the approximate roots in the (*p*, *q*)-sine Bernoulli polynomials. The conditions are confirmed by equating them to the conditions of the (*p*, *q*)-cosine Bernoulli polynomials. The stacking structure of the approximate roots of the (*p*, *q*)-sine Bernoulli polynomials when *p* = 0.5, *q* = 0.9, and *y* = 5 can be checked in Figure 3. At 1 ≤ *n* ≤ 30, the distribution range of the approximate roots appears wider in the values on the real axis than in the imaginary axis, as shown in the left picture in Figure 3. Figure 3 reveals that, as the value of *n* becomes larger, the approximate roots become more symmetric, and the approximate form approaches a circular shape, including the origin.

**Figure 3.** Stacking structure of approximate roots in the (*p*, *q*)-sine Bernoulli polynomials when *p* = 0.5, *q* = 0.9, and *y* = 5 in 3D.

When we change the value of *p*, the structure of the approximate roots of the (*p*, *q*)-sine Bernoulli polynomials when *p* = 0.1 under the same conditions as in Figure 3 is presented in Figure 4. In comparison with Figure 3, the area of the real and the imaginary axes in Figure 4 is greater, and the approximate roots have a wider distribution than observed in Figure 3. This property is common in the approximate roots of the (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials. This indicates that, as the value of *p* decreases, the approximate roots of the (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials spread wider. In addition, as displayed in Figure 4, the structure of the approximate roots of the (*p*, *q*)-sine Bernoulli polynomials is symmetric as the value of *n* increases.

**Figure 4.** Stacking structure of approximate roots in the (*p*, *q*)-sine Bernoulli polynomials when *p* = 0.1, *q* = 0.9, and *y* = 5 in 3D.

### **5. Conclusions and Future Directions**

In this paper, we explained about the (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials, their basic properties, and various symmetric properties. Based on the above contents, we identified the structures of the approximate roots of the (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials. As a result, we observed that the above polynomials obtain a structure of approximate roots, which has certain patterns and has a symmetric property under the given circumstances.

Further study is needed regarding whether the structure of approximate roots for the (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials have symmetric properties under different circumstances. Furthermore, we think researching theories related to this topic is important to mathematicians.

**Author Contributions:** Conceptualization, J.Y.K.; Methodology, C.S.R.; Writing—original draft, J.Y.K. These autors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

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

**Acknowledgments:** The authors would like to thank the referees for their valuable comments, which improved the original manuscript in its present form.

**Conflicts of Interest:** The authors declare that there is no conflict of interests regarding the publication of this paper.
