**Proof.**

(i) Suppose that *ep*,*<sup>q</sup>*(*t*) = 1 in the generating function of the (*p*, *q*)-cosine Bernoulli polynomials. Then, we have

$$\sum\_{n=0}^{\infty} \, \_C B\_{n,p,\emptyset}(x,y) \frac{t^n}{[n]\_{p,\emptyset}!} (e\_{p,\emptyset}(t) - 1) = t e\_{p,\emptyset}(t\mathbf{x}) \mathbf{CO} S\_{p,\emptyset}(t\mathbf{y}).\tag{33}$$

We write the left-hand side of Equation (33) as follows:

$$\begin{split} &\sum\_{n=0}^{\infty} \, \_CB\_{n,p,q}(x,y) \frac{t^n}{[n]\_{p,q}!} (\varepsilon\_{p,q}(t) - 1) \\ &= \sum\_{n=0}^{\infty} \, \_CB\_{n,p,q}(x,y) \frac{t^n}{[n]\_{p,q}!} \left( \sum\_{n=0}^{\infty} p^{\left(\frac{n}{2}\right)} \frac{t^n}{[n]\_{p,q}!} - 1 \right) \\ &= \sum\_{n=0}^{\infty} \left( \sum\_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} p^{\binom{n-k}{2}} \, \_CB\_{k,p,q}(x,y) - \, \_CB\_{n,p,q}(x,y) \right) \frac{t^n}{[n]\_{p,q}!} \, \_CB\_{n,p,q}(x,y) \end{split} \tag{34}$$

and we transform the right-hand side into the following:

$$\begin{split} \text{tr}\_{p,q}(\text{tx}) \text{COS}\_{p,q}(\text{ty}) &= \sum\_{n=0}^{\infty} \text{C}\_{n,p,q}(\text{x}, \text{y}) \frac{t^{n+1}}{[n]\_{p,q}!} \\ &= \sum\_{n=0}^{\infty} [n]\_{p,q} \text{C}\_{n-1,p,q}(\text{x}, \text{y}) \frac{t^{n}}{[n]\_{p,q}!} . \end{split} \tag{35}$$

Therefore, we obtain the following:

$$\sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} p^{\binom{n-k}{2}} \subset \mathcal{B}\_{k,p,q}(\mathbf{x}, \mathbf{y}) - \subset \mathcal{B}\_{n,p,q}(\mathbf{x}, \mathbf{y}) = [n]\_{p,q} \mathbb{C}\_{n-1,p,q}(\mathbf{x}, \mathbf{y}).\tag{36}$$

By calculating the left-hand side of Equation (36), we investigate the required result.

(ii) We do not include the proof of Theorem 5 (*ii*) because the proving process is similar to that of Theorem 5 (*i*).

**Corollary 2.** *Setting p* = 1 *in Theorem 5, the following equations hold*

$$\begin{aligned} (i) \quad [n]\_{\emptyset} \mathbb{C}\_{n-1, \emptyset}(\mathbf{x}, \mathbf{y}) &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{\emptyset} \mathbb{C} B\_{k, \emptyset}(\mathbf{x}, \mathbf{y}) - \mathbb{C} B\_{n, \emptyset}(\mathbf{x}, \mathbf{y}) \\\ (ii) \quad [n]\_{\emptyset} \mathbb{S}\_{n-1, \emptyset}(\mathbf{x}, \mathbf{y}) &= \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{\emptyset} \mathbb{S}\_{k, \emptyset}(\mathbf{x}, \mathbf{y}) - {}\_{\emptyset} B\_{n, \emptyset}(\mathbf{x}, \mathbf{y}), \end{aligned} \tag{37}$$

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

**Corollary 3.** *Assigning p* = 1 *and q* → 1 *in Theorem 5, the following holds:*

$$\begin{aligned} (i) \quad n \mathbb{C}\_{n-1}(\mathbf{x}, \mathbf{y}) &= \sum\_{k=0}^{n-1} \binom{n}{k} \, \_\text{C}B\_n(\mathbf{x}, \mathbf{y})\\ (ii) \quad n \mathbb{S}\_{n-1}(\mathbf{x}, \mathbf{y}) &= \sum\_{k=0}^{n-1} \binom{n}{k} \, \_\text{S}B\_n(\mathbf{x}, \mathbf{y}) \end{aligned} \tag{38}$$

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

**Theorem 6.** *Let* |*p*/*q*| < 1*. Then, we have*

$$\begin{split} (i) \quad & \, \_C B\_{n,p,q}(1,y) = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} (-1)^{n-k} q^{\binom{n-k}{2}} \left( [k]\_{p,q} \mathbb{C}\_{k-1,p,q}(\mathbf{x},y) + \_C B\_{k,p,q}(\mathbf{x},y) \right) \mathbf{x}^{n-k}, \\ (ii) \quad & \, \_S B\_{n,p,q}(1,y) = \sum\_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}\_{p,q} (-1)^{n-k} q^{\binom{n-k}{2}} \left( [k]\_{p,q} \mathbb{S}\_{k-1,p,q}(\mathbf{x},y) + \_S B\_{k,p,q}(\mathbf{x},y) \right) \mathbf{x}^{n-k}. \end{split} \tag{39}$$
