**1. Introduction**

In 1991, (*p*, *q*)-calculus including (*p*, *q*)-number with two independent variables *p* and *q*, was first independently considered [1,2]. Throughout this paper, the sets of natural numbers, integers, real numbers and complex numbers are denoted by N,Z, R and C, respectively.

For any *n* ∈ N, the (*p*, *q*)-number is defined by the following:

$$[n]\_{p,q} = \frac{p^n - q^n}{p - q}, \qquad \text{where} \quad |p/q| < 1,\tag{1}$$

which is a natural generalization of the *q*-number. From Equation (1), we note that [*n*]*<sup>p</sup>*,*<sup>q</sup>* = [*n*]*<sup>q</sup>*,*p*.

Many physical and mathematical problems have led to the necessity of studying (*p*, *q*)-calculus. Since 1991, many mathematicians and physicists have developed (*p*, *q*)-calculus in several different research areas. For example, in 1994, [3] introduced (*p*, *q*)-hypergeometric functions. Three years later, [3,4] derived related preliminary results by considering a more general (*p*, *q*)-hypergeometric series and Burban's (*p*, *q*)-hypergeometric series, respectively. In 2005, based on the (*p*, *q*)-numbers, [5] studied about (*p*, *q*)-hypergeometric series and discovered results corresponding to the (*p*, *q*)-extensions of known *q*-identities. Moreover, [6] established properties similar to the ordinary and *q*-binomial coefficients after developing the (*p*, *q*)-hypergeometric series in 2008. About seven years later, [7] introduced (*p*, *q*)-gamma and (*p*, *q*)-beta functions, which are generalizations of the gamma and beta functions.

The different variations of Bernoulli polynomials, *q*-Bernoulli polynomials and (*p*, *q*)-Bernoulli polynomials are illustrated in the diagram below. Kim, Ryoo and many mathematicians have studied the first and second rows of the polynomials in the diagram(see [8–12]). These studies began producing valuable results in areas related to number theory and combinatorics.

The main idea is to use property of (*p*, *q*)-numbers and combine (*p*, *q*)-trigonometric functions. From this idea, we construct (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials. Investigating the various explicit identities for (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials in the diagram's third row is the main goal of this paper.

$$\begin{array}{c} \frac{t}{\varepsilon^{\varepsilon}-1}e^{tx} = \sum\_{n=0}^{\infty}B\_{n}(x)\frac{t}{n!} \qquad \qquad \qquad \qquad \qquad \frac{t}{\varepsilon^{\varepsilon}-1}e^{tx}\cos(ty) = \sum\_{n=0}^{\infty}B\_{n}^{(\varepsilon)}(x,y)\frac{t^{n}}{n!} \\ \text{(Bernoulli polynomials)} \qquad \qquad \qquad \qquad \frac{t}{\varepsilon^{\varepsilon}-1}e^{tx}\sin(ty) = \sum\_{n=0}^{\infty}B\_{n}^{(\varepsilon)}(x,y)\frac{t^{n}}{n!} \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \frac{t}{\varepsilon^{\varepsilon}-1}e^{tx}\sin(ty) = \sum\_{n=0}^{\infty}B\_{n}^{(\varepsilon)}(x,y)\frac{t^{n}}{n!} \\ \text{(\*)} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left\{ \begin{array}{c} \frac{t}{\varepsilon^{\varepsilon}-1}e^{tx}\sin(ty) = \sum\_{n=0}^{\infty}B\_{n}^{(\varepsilon)}(x,y)\frac{t^{n}}{n!} \\ \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left\{ \begin{array}{c} \frac{t}{\varepsilon^{\varepsilon}-1}e^{tx}\sin(ty) = \sum\_{n=0}^{\infty}B\_{n}^{(\varepsilon)}(x,y) = \sum\_{n=0}^{\infty}B\_{n}^{(\varepsilon)}(x,y)\frac{t^{n}}{n!} \end{array} \right. \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left\{ \begin{array}{c} \frac{t}{\varepsilon q^{\varepsilon}}\cos(ty) = \sum\_{n=0}^{\infty}B\_{n}(x,y)\frac{t^{n}}{n!} \\ \quad \qquad \qquad \qquad \qquad \qquad \qquad \$$

Due to their importance, the classical Bernoulli, Euler, and Genocchi polynomials have been studied extensively and are well-known. Mathematicians have studied these polynomials through various mathematical applications including finite difference calculus, *p*-adic analytic number theory, combinatorial analysis and number theory. Many mathematicians are familiar with the theorems and definitions of classical Bernoulli, Euler, and Genocchi polynomials. Based on the theorems and definitions, it is significant to study these properties in various ways by the combining with Bernoulli, Euler, and Genocchi polynomials. Mathematicians are studying the extended versions of these polynomials and are researching new polynomials by combining mathematics with other fields, such as physics or engineering (see [9–14]). The definition of Bernoulli polynomials combined with (*p*, *q*)-numbers follows:

**Definition 1.** *The* (*p*, *q*)*-Bernoulli numbers, Bn*,*p*,*q, and polynomials, Bn*,*p*,*<sup>q</sup>*(*z*)*, can be expressed as follows (see [8])*

$$\sum\_{n=0}^{\infty} B\_{n,p,q} \frac{t^n}{[n]\_{p,q}!} = \frac{t}{\varepsilon\_{p,q}(t) - 1}, \quad \sum\_{n=0}^{\infty} B\_{n,p,q}(z) \frac{t^n}{[n]\_{p,q}!} = \frac{t}{\varepsilon\_{p,q}(t) - 1} \varepsilon\_{p,q}(tz). \tag{2}$$

In [11], we confirmed the properties of *q*-cosine and *q*-sine Bernoulli polynomials. Their definitions and representative properties are as follows.

**Definition 2.** *The q-cosine Bernoulli polynomials <sup>C</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *and q-sine Bernoulli polynomials <sup>S</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *are defined by the following:*

$$\sum\_{n=0}^{\infty} \, \_CB\_{n,q}(\mathbf{x}, y) \frac{t^n}{n!} = \frac{t}{\varepsilon\_q(t) - 1} \varepsilon\_q(t \mathbf{x}) \text{COS}\_q(\mathbf{t} y), \quad \sum\_{n=0}^{\infty} \, \_SB\_n(\mathbf{x}, y) \frac{t^n}{n!} = \frac{t}{\varepsilon\_q(t) - 1} \varepsilon\_q(t \mathbf{x}) \text{SIN}\_q(\mathbf{t} y). \tag{3}$$

**Theorem 1.** *For x*, *y* ∈ R*, we have the following:*

*(i)* ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *<sup>C</sup>Bn*,*<sup>q</sup>*((*<sup>x</sup>* ⊕ *<sup>r</sup>*)*<sup>q</sup>*, *y*) + *<sup>S</sup>Bn*,*<sup>q</sup>*((*<sup>x</sup> <sup>r</sup>*)*<sup>q</sup>*, *y*) = ∑*n k*=0 % *n k* & *q q*( *<sup>n</sup>*−*k* 2 )*rn*−*<sup>k</sup> <sup>C</sup>Bk*,*<sup>q</sup>*(*<sup>x</sup>*, *y*)+( −<sup>1</sup>)*<sup>n</sup>*−*<sup>k</sup> <sup>S</sup>Bk*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *<sup>S</sup>Bn*,*<sup>q</sup>*((*<sup>x</sup>* ⊕ *<sup>r</sup>*)*<sup>q</sup>*, *y*) + *<sup>C</sup>Bn*,*<sup>q</sup>*((*<sup>x</sup> <sup>r</sup>*)*<sup>q</sup>*, *y*) = ∑*n k*=0 % *n k* & *q q*( *<sup>n</sup>*−*k* 2 )*rn*−*<sup>k</sup> <sup>S</sup>Bk*,*<sup>q</sup>*(*<sup>x</sup>*, *y*)+( −<sup>1</sup>)*<sup>n</sup>*−*<sup>k</sup> <sup>C</sup>Bk*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *(ii)* - *∂ ∂x <sup>C</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *<sup>y</sup>*)=[*n*]*qCBn*−1,*q*(*<sup>x</sup>*, *y*), *∂ ∂y <sup>C</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) = <sup>−</sup>[*n*]*qSBn*−1,*q*(*<sup>x</sup>*, *qy*) *∂ ∂x <sup>S</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *<sup>y</sup>*)=[*n*]*qSBn*−1,*q*(*<sup>x</sup>*, *y*), *∂ ∂y <sup>S</sup>Bn*,*<sup>q</sup>*(*<sup>x</sup>*, *<sup>y</sup>*)=[*n*]*qCBn*−1,*q*(*<sup>x</sup>*, *qy*) (4)

The main goal of this paper is to identify the properties of (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials. In Section 2, we review some definitions and theorem of (*p*, *q*)-calculus. In Section 3, we introduce (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials. Using the properties of exponential functions and trigonometric functions associated with (*p*, *q*)-numbers, we determine the various properties and identities of the polynomials. Section 4 presents the investigation of the symmetric properties of (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials in different forms and based on the symmetric polynomials, we check the symmetric structure of the approximate roots.
