**2. Preliminaries**

In this section, we introduce definitions and preliminary facts that are used throughout this paper (see [6,12–20]).

**Definition 3.** *For n* ≥ *k, the Gaussian binomial coefficients are defined by the following:*

$$
\begin{bmatrix} m \\ r \end{bmatrix}\_{p,q} = \frac{[n]\_{p,q}!}{[n-k]\_{p,q}! [k]\_{p,q}!} \tag{5}
$$

*where m and r are non-negative integers.*

We note that [*n*]*<sup>p</sup>*,*q*! = [*n*]*<sup>p</sup>*,*<sup>q</sup>*[*<sup>n</sup>* − <sup>1</sup>]*<sup>p</sup>*,*<sup>q</sup>* ··· [2]*<sup>p</sup>*,*<sup>q</sup>*[1]*<sup>p</sup>*,*q*, where *n* ∈ N. For *r* = 0, the value of the equation is 1, because both the numerator and denominator are empty products. Moreover, (*p*, *q*)-analogues of the binomial formula exist, and this definition has numerous properties.

**Definition 4.** *The* (*p*, *q*)*-analogues of* (*x* − *a*)*<sup>n</sup> and* (*x* + *a*)*<sup>n</sup> are defined by the following:*

$$\begin{split} \text{(i)} \quad & (\mathbf{x} \odot \mathbf{a})\_{p,q}^{\mathrm{n}} = \begin{cases} 1, & \text{if } n=0\\ (\mathbf{x}-a)(p\mathbf{x}-q\mathbf{a}) \cdot \cdots (p^{n-1}\mathbf{x}-q^{n-1}a), & \text{if } n \ge 1 \end{cases} \\ \text{(ii)} \ (\mathbf{x} \oplus a)\_{p,q}^{\mathrm{n}} = \begin{cases} 1, & \text{if } n=0\\ (\mathbf{x}+a)(p\mathbf{x}+q\mathbf{a}) \cdot \cdots (p^{n-2}\mathbf{x}+q^{n-2}a)(p^{n-1}\mathbf{x}+q^{n-1}a), & \text{if } n \ge 1 \end{cases} \\ &= \sum\_{k=0}^{n} \begin{bmatrix} n\\ k \end{bmatrix}\_{p,q} p^{(\frac{k}{2})} q^{n^{2}-k}, \end{split} \tag{6}$$

**Definition 5.** *We express the two forms of* (*p*, *q*)*-exponential functions as follows:*

$$c\_{p,q}(\mathbf{x}) = \sum\_{n=0}^{\infty} p^{\binom{n}{2}} \frac{\mathbf{x}^n}{[n]\_{p,q}!}, \qquad E\_{p,q}(\mathbf{x}) = \sum\_{n=0}^{\infty} q^{\binom{n}{2}} \frac{t^n}{[n]\_{p,q}!}. \tag{7}$$

From Equation (7), we determine an important property, *ep*,*<sup>q</sup>*(*x*)*Ep*,*<sup>q</sup>*(−*<sup>x</sup>*) = 1. Moreover, Duran, Acikgos, and Araci defined '*ep*,*<sup>q</sup>*(*x*) in [17] as follows:

$$\widetilde{\epsilon}\_{p,q}(\mathbf{x}) = \sum\_{n=0}^{\infty} \frac{\mathbf{x}^n}{[n]\_{p,q}!}. \tag{8}$$

From Equations (8) and (6), we remark *ep*,*<sup>q</sup>*(*x*)*Ep*,*<sup>q</sup>*(*y*) = '*ep*,*<sup>q</sup>*(*<sup>x</sup>* ⊕ *<sup>y</sup>*)*<sup>p</sup>*,*q*.

**Definition 6.** *For x* = 0*, the* (*p*, *q*)*-derivative of a function f with respect to x is defined by the following:*

$$D\_{p,q}f(\mathbf{x}) = \frac{f(p\mathbf{x}) - f(q\mathbf{x})}{(p-q)\mathbf{x}},\tag{9}$$

where (*Dp*,*<sup>q</sup> f*)(0) = *f* (0), which prove that *f* is differentiable at 0. Moreover, it is evident that *Dp*,*qx<sup>n</sup>* = [*n*]*<sup>p</sup>*,*qxn*−1.

**Definition 7.** *Let i* = √−1 ∈ C*. Then the* (*p*, *q*)*-trigonometric functions are defined by the following:*

$$\begin{aligned} \sin\_{p,q}(\mathbf{x}) &= \frac{\varepsilon\_{p,q}(i\mathbf{x}) - \varepsilon\_{p,q}(-i\mathbf{x})}{2i}, & \text{SIN}\_{p,q}(\mathbf{x}) &= \frac{E\_{p,q}(i\mathbf{x}) - E\_{p,q}(-i\mathbf{x})}{2i} \\ \cos\_{p,q}(\mathbf{x}) &= \frac{\varepsilon\_{p,q}(i\mathbf{x}) + \varepsilon\_{p,q}(-i\mathbf{x})}{2}, & \text{COS}\_{p,q}(\mathbf{x}) &= \frac{E\_{p,q}(i\mathbf{x}) + E\_{p,q}(-i\mathbf{x})}{2}, \end{aligned} \tag{10}$$

*where, SINp*,*<sup>q</sup>*(*x*) = *sinp*−1,*q*−<sup>1</sup> (*x*) *and COSp*,*<sup>q</sup>*(*x*) = *cosp*−1,*q*−<sup>1</sup> (*x*)*.*

In the same way as well-known Euler expressions using exponential functions, we define the (*p*, *q*)-analogues of hyperbolic functions and find several formulae (see [3,5,17]).

**Theorem 2.** *The following relationships hold:*

$$\begin{aligned} \text{(i)} \quad &\sin\_{p,q}(\mathbf{x}) \text{COS}\_{p,q}(\mathbf{x}) = \cos\_{p,q}(\mathbf{x}) \text{SIN}\_{p,q}(\mathbf{x})\\ \text{(ii)} \quad &e\_{p,q}(\mathbf{x}) = \cosh\_{p,q}(\mathbf{x}) \sin \text{h}\_{p,q}(\mathbf{x})\\ \text{(iii)} \quad &E\_{p,q}(\mathbf{x}) = \text{COSH}\_{p,q}(\mathbf{x}) \text{SINH}\_{p,q}(\mathbf{x}). \end{aligned} \tag{11}$$

From Definition 7 and Theorem 2, we note that cosh*p*,*<sup>q</sup>*(*x*)COSH*p*,*<sup>q</sup>*(*x*) − sinh*p*,*<sup>q</sup>*(*x*) SINH*p*,*<sup>q</sup>*(*x*) = 1.

### **3. Several Basic Properties of (***p***,** *q***)-Cosine and (***p***,** *q***)-Sine Bernoulli Polynomials**

We look for Lemma 1 and Theorem 3 in order to introduce (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials. From the definitions of the (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials, we search for a variety of properties. We also find relationships with other polynomials using properties of (*p*, *q*)-trigonometric functions or other methods.

**Lemma 1.** *For y* ∈ R *and i* = √−<sup>1</sup>*, we have the following:*

$$\begin{aligned} (i) \quad & E\_{p,q}(\text{ity}) = \text{COS}\_{p,q}(\text{ty}) + i \text{SIN}\_{p,q}(\text{ty}),\\ (ii) \quad & E\_{p,q}(-\text{ity}) = \text{COS}\_{p,q}(\text{ty}) - i \text{SIN}\_{p,q}(\text{ty}). \end{aligned} \tag{12}$$

**Proof.**

(i) *Ep*,*<sup>q</sup>*(*ity*) can be expressed using the (*p*, *q*)-cosine and (*p*, *q*)-sine functions as

$$\begin{split} E\_{p,q}(\text{ity}) &= \frac{E\_{p,q}(\text{ity}) + E\_{p,q}(-\text{ity})}{2} + \frac{E\_{p,q}(\text{ity}) - E\_{p,q}(-\text{ity})}{2} \\ &= \text{COS}\_{p,q}(\text{ty}) + iSIN\_{p,q}(\text{ty}). \end{split} \tag{13}$$

(ii) By substituting −*ity* instead of (*i*), we obtain the following:

$$\begin{split} E\_{p,\emptyset}(-\text{ity}) &= \frac{E\_{p,\emptyset}(-\text{ity}) + E\_{p,\emptyset}(-\text{ity})}{2} - \frac{E\_{p,\emptyset}(-\text{ity}) - E\_{p,\emptyset}(-\text{ity})}{2} \\ &= \text{COS}\_{p,\emptyset}(\text{ty}) - iSIN\_{p,\emptyset}(\text{ty}). \end{split} \tag{14}$$

Therefore, we complete the proof of Lemma 1.

We note the following relations between *ep*,*q*, *Ep*,*<sup>q</sup>* and '*ep*,*q*.

$$\begin{aligned} (i) \quad & \varepsilon\_{p,q}(\mathbf{x}) E\_{p,q}(\mathbf{y}) = \sum\_{n=0}^{\infty} \frac{(\mathbf{x} \oplus \mathbf{y})\_{p,q}^n}{[n]\_{p,q}!} = \widetilde{\varepsilon}\_{p,q}(\mathbf{x} \oplus \mathbf{y})\_{p,q}, \\\ (ii) \quad & \varepsilon\_{p,q}(\mathbf{x}) E\_{p,q}(-\mathbf{y}) = \sum\_{n=0}^{\infty} \frac{(\mathbf{x} \ominus \mathbf{y})\_{p,q}^n}{[n]\_{p,q}!} = \widetilde{\varepsilon}\_{p,q}(\mathbf{x} \ominus \mathbf{y})\_{p,q}. \end{aligned} \tag{15}$$

**Theorem 3.** *Let x*, *y* ∈ R*, i* = √−<sup>1</sup>*, and* |*q*/*p*| < 1*. Then, we have*

$$\begin{split} (i) \quad & \sum\_{n=0}^{\infty} \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \left( \frac{(\mathbf{x} \oplus iy)\_{p,q}^{k} + (\mathbf{x} \odot iy)\_{p,q}^{k}}{2} \right) B\_{n-k,p,q} \frac{t^{n}}{[n]\_{p,q}!} \\ & = \frac{t}{\varepsilon\_{p,q}(t) - 1} c\_{p,q}(t\mathbf{x}) \mathbf{C} \mathbf{O}\_{p,q}(ty), \\ (ii) \quad & \sum\_{n=0}^{\infty} \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \left( \frac{(\mathbf{x} \oplus iy)\_{p,q}^{k} - (\mathbf{x} \odot iy)\_{p,q}^{k}}{2i} \right) B\_{n-k,p,q} \frac{t^{n}}{[n]\_{p,q}!} \\ & = \frac{t}{\varepsilon\_{p,q}(t) - 1} c\_{p,q}(t\mathbf{x}) S \mathbf{N}\_{p,q}(ty). \end{split} \tag{16}$$

**Proof.**

(i) We note that

$$\sum\_{n=0}^{\infty} B\_{n,p,q} \frac{t^n}{[n]\_{p,q}!} = \frac{t}{\varepsilon\_{p,q}(t) - 1}. \tag{17}$$

We find the following by multiplying '*ep*,*<sup>q</sup> t*(*x* ⊕ *<sup>y</sup>*)*<sup>p</sup>*,*<sup>q</sup>* in both sides of Equation (17).

$$\sum\_{n=0}^{\infty} B\_{n,p,q} \frac{t^n}{[n]\_{p,q}!} \check{\varepsilon}\_{p,q} \left( t(\mathbf{x} \oplus y)\_{p,q} \right) = \frac{t}{\varepsilon\_{p,q}(t) - 1} \check{\varepsilon}\_{p,q} \left( t(\mathbf{x} \oplus y)\_{p,q} \right). \tag{18}$$

The left-hand side of Equation (18) can be changed into

$$\begin{split} \sum\_{n=0}^{\infty} B\_{n,p,q} \frac{t^n}{[n]\_{p,q}!} \tilde{\epsilon}\_{p,q} \left( \mathbf{t} \left( \mathbf{x} \oplus y \right)\_{p,q} \right) &= \sum\_{n=0}^{\infty} B\_{n,p,q} \frac{t^n}{[n]\_{p,q}!} \sum\_{n=0}^{\infty} \left( \mathbf{x} \oplus y \right)\_{p,q}^n \frac{t^n}{[n]\_{p,q}!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} (\mathbf{x} \oplus y)\_{p,q}^k B\_{n-k,p,q} \right) \frac{t^n}{[n]\_{p,q}!} \end{split} \tag{19}$$

and by using Lemma 1 (*i*) on the right-hand side of Equation (18), we yield

$$\begin{split} \frac{t}{e\_{p,q}(t) - 1} \tilde{\varepsilon}\_{p,q} \left( t(\mathbf{x} \oplus \boldsymbol{y})\_{p,q} \right) &= \frac{t}{e\_{p,q}(t) - 1} \varepsilon\_{p,q}(\mathbf{x}) E\_{p,q}(\boldsymbol{y}) \\ &= \frac{t e\_{p,q}(\mathbf{x})}{e\_{p,q}(t) - 1} \left( \text{CO}\_{p,q}(t\boldsymbol{y}) + i S \text{IN}\_{p,q}(t\boldsymbol{y}) \right) . \end{split} \tag{20}$$

From Equations (19) and (20), we derive the following:

$$\sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} (\mathbf{x} \oplus y)\_{p,q}^k B\_{n-k,p,q} \right) \frac{t^n}{[n]\_{p,q}!} = \frac{t e\_{p,q}(\mathbf{x})}{\varepsilon\_{p,q}(t) - 1} \left( \text{COS}\_{p,q}(\mathbf{t}y) + i \text{SIN}\_{p,q}(\mathbf{t}y) \right). \tag{21}$$

We obtain the equation below for (*p*, *q*)-Bernoulli numbers using a similar method.

$$\sum\_{n=0}^{\infty} \left( \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} (\mathbf{x} \odot i\mathbf{y})\_{p,q}^k B\_{n-k,p,q} \right) \frac{t^n}{[n]\_{p,q}!} = \frac{t e\_{p,q}(t\mathbf{x})}{e\_{p,q}(t) - 1} \left( \mathbb{C} \mathbf{O} S\_{p,q}(t\mathbf{y}) - i \mathbb{S} \mathbf{I} N\_{p,q}(t\mathbf{y}) \right). \tag{22}$$

By using Equations (21) and (22), we have

$$\sum\_{n=0}^{\infty} \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \left( \frac{(\mathbf{x} \ominus i\mathbf{y})\_{p,q}^{k} + (\mathbf{x} \ominus i\mathbf{y})\_{p,q}^{k}}{2} \right) B\_{n-k,p,q} \frac{t^{n}}{[n]\_{p,q}!} = \frac{t}{c\_{p,q}(t) - 1} \epsilon\_{p,q}(t\mathbf{x}) \text{COS}\_{p,q}(t\mathbf{y}) \tag{23}$$

and

$$\sum\_{n=0}^{\infty} \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} \left( \frac{(\mathbf{x} \ominus i\mathbf{y})\_{p,q}^{k} - (\mathbf{x} \ominus i\mathbf{y})\_{p,q}^{k}}{2i} \right) B\_{n-k,p,q} \frac{t^{n}}{[n]\_{p,q}!} = \frac{t}{\varepsilon\_{p,q}(t) - 1} \epsilon\_{p,q}(t\mathbf{x}) SIN\_{p,q}(t\mathbf{y}).\tag{24}$$

Therefore, we can conclude the required results.

Thus, we are ready to introduce (*p*, *q*)-cosine and (*p*, *q*)-sine Bernoulli polynomials using Lemma 1 and Theorem 3.

**Definition 8.** *Let* |*p*/*q*| < 1 *and x*, *y* ∈ R*. Then* (*p*, *q*)*-cosine and* (*p*, *q*)*-sine Bernoulli polynomials are respectively defined by the following:*

$$\sum\_{n=0}^{\infty} \, \_CB\_{n,p,q}(x,y) \frac{t^n}{[n]\_{p,q}!} = \frac{t}{\varepsilon\_{p,q}(t) - 1} \varepsilon\_{p,q}(tx) \, \text{CO}\_{p,q}(ty),$$

$$\sum\_{n=0}^{\infty} \, \_SB\_{n,p,q}(x,y) \frac{t^n}{[n]\_{p,q}!} = \frac{t}{\varepsilon\_{p,q}(t) - 1} \varepsilon\_{p,q}(tx) \, \text{SIN}\_{p,q}(ty). \tag{25}$$

*and*

$$\text{From Definition 8, we determine } q\text{-cosine and } q\text{-sine Bernoulli polynomials when } |q| < 1 \text{ and } p = 1. \text{ In addition, we observe cosine Bernoulli polynomials and since Bernoulli polynomials for } q \to 1, \text{ and } p = 1.$$

**Corollary 1.** *From Theorem 3 and Definition 8, the following holds*

$$\begin{aligned} (i) \quad & \, \_C B\_{n, p, q}(x, y) = \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p, q} \left( \frac{(\mathbf{x} \odot i\mathbf{y})\_{p, q}^k + (\mathbf{x} \odot i\mathbf{y})\_{p, q}^k}{2} \right) B\_{n-k, p, q}, \\ (ii) \, \_S B\_{n, p, q}(\mathbf{x}, y) &= \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p, q} \left( \frac{(\mathbf{x} \odot i\mathbf{y})\_{p, q}^k - (\mathbf{x} \odot i\mathbf{y})\_{p, q}^k}{2i} \right) B\_{n-k, p, q}. \end{aligned} \tag{26}$$

*where Bn*,*p*,*<sup>q</sup> denotes the* (*p*, *q*)*-Bernoulli numbers.*

**Example 1.** *From Definition 8, a few examples of <sup>C</sup>Bn*,*p*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *and <sup>S</sup>Bn*,*p*,*<sup>q</sup>*(*<sup>x</sup>*, *y*) *are the follows:*

$$\begin{aligned} \{\Box\_{B,p,q}(\mathbf{x},\mathbf{y}) &= 0\\ \Box\_{B,p,q}(\mathbf{x},\mathbf{y}) &= p\mathbf{x} \\ \Box\_{B,p,q}(\mathbf{x},\mathbf{y}) &= p^2\mathbf{x}^2 - q\mathbf{y}^2 \\ \Box\_{B,p,q}(\mathbf{x},\mathbf{y}) &= p^3\mathbf{x}^3 - pq(p^2 + pq + q^2)\mathbf{x}\mathbf{y}^2 \\ \Box\_{B,p,q}(\mathbf{x},\mathbf{y}) &= p^4\mathbf{x}^4 - p^2q(p^2 + q^2)(p^2 + pq + q^2)\mathbf{x}^2\mathbf{y}^2 + q^6\mathbf{y}^4, \\ \vdots &\dots \end{aligned} \tag{27}$$

*and*

$$\begin{aligned} \, \_SB\_{1,p,q}(x,y) &= 0\\ \, \_SB\_{1,p,q}(x,y) &= \frac{y}{p+q}\\ \, \_SB\_{2,p,q}(x,y) &= \frac{pxy}{p^2+pq+q^2} \\ \, \_SB\_{3,p,q}(x,y) &= \frac{y\left(\frac{p^2x^2}{p^2+q^2}-q^3y^2\right)}{p+q} \\ \, \_SB\_{4,p,q}(x,y) &= p(p-q)xy\left(\frac{p^2x^2}{p^5-q^5}+\frac{q^3(p^2+q^2)y^2}{-p^3+q^3}\right), \\ &\quad \times = \dots \end{aligned} \tag{28}$$

**Definition 9.** *Let* |*p*/*q*| < 1*. Then, we define*

$$\sum\_{n=0}^{\infty} \mathbb{C}\_{n,p,q}(\mathbf{x}, \mathbf{y}) \frac{t^n}{[n]\_{p,q}!} = \varepsilon\_{p,q}(t\mathbf{x}) \text{COS}\_{p,q}(t\mathbf{y}), \quad \sum\_{n=0}^{\infty} \text{S}\_{n,p,q}(\mathbf{x}, \mathbf{y}) \frac{t^n}{[n]\_{p,q}!} = \varepsilon\_{p,q}(t\mathbf{x}) \text{S} \text{IN}\_{p,q}(t\mathbf{y}). \tag{29}$$

**Theorem 4.** *Let k be a nonnegative integer and* |*p*/*q*| < 1*. Then, we have*

$$\begin{aligned} (i) \quad & \, \_C B\_{n,p,q}(\mathbf{x}, \mathbf{y}) = \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} B\_{n-k,p,q} \mathbf{C}\_{k,p,q}(\mathbf{x}, \mathbf{y}),\\ (ii) \quad & \, \_S B\_{n,p,q}(\mathbf{x}, \mathbf{y}) = \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} B\_{n-k,p,q} \mathbf{S}\_{k,p,q}(\mathbf{x}, \mathbf{y}),\end{aligned} \tag{30}$$

*where Bn*,*p*,*<sup>q</sup> is the* (*p*, *q*)*-Bernoulli numbers.* **Proof.**

(i) Using the generating function of the (*p*, *q*)-cosine Bernoulli polynomials and Definition 9, we find

$$\begin{split} \sum\_{n=0}^{\infty} \, \_C B\_{n,p,q}(x,y) \frac{t^n}{[n]\_{p,q}!} &= \sum\_{n=0}^{\infty} B\_{n,p,q} \frac{t^n}{[n]\_{p,q}!} \sum\_{n=0}^{\infty} \, \_C\mathbb{C}\_{n,p,q}(x,y) \frac{t^n}{[n]\_{p,q}!} \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} B\_{n-k,p,q} \mathbb{C}\_{n,p,q}(x,y) \right) \frac{t^n}{[n]\_{p,q}!} . \end{split} \tag{31}$$

Through comparison of the coefficients of both sides for Equation (31), we obtain the desired results immediately.

(ii) By applying a method similar to (*i*) in the generating function of the (*p*, *q*)-sine Bernoulli polynomials, we complete the proof of Theorem 4 (*ii*).

**Theorem 5.** *For a nonnegative integer n, we derive*

$$\begin{aligned} (i) \quad [n]\_{p,q} \mathbb{C}\_{n-1,p,q}(\mathbf{x}, \mathbf{y}) &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} p^{(\frac{n-k}{2})} \circ B\_{k,p,q}(\mathbf{x}, \mathbf{y}) - \mathbb{C}B\_{n,p,q}(\mathbf{x}, \mathbf{y}), \\\ (ii) \quad [n]\_{p,q} \mathbb{S}\_{n-1,p,q}(\mathbf{x}, \mathbf{y}) &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} p^{(\frac{n-k}{2})} \circ B\_{k,p,q}(\mathbf{x}, \mathbf{y}) - \mathbb{S}B\_{n,p,q}(\mathbf{x}, \mathbf{y}). \end{aligned} \tag{32}$$
