A Note on the (p, q)-Derivative Operator

In this note, for the (p, q)-derivative operator \(D_{p, q}\) $D_{p,q}$ defined by \(\begin{aligned}D_{p, q}f(x)={\left\{ \begin{array}{ll} \frac{f(px)-f(qx)}{(p-q)x} & \text {if } x\ne 0;\\ f'(0), & \text {if}~x=0. \end{array}\right. }, \quad \text { where } p\ne q,\end{aligned}\) $$\begin{array}{c}\hfill D_{p,q}f(x)=\left\{\begin{array}{cc}\frac{f(px)-f(qx)}{(p-q)x}\hfill & \text{if}x\ne 0;\hfill \\ f^{\prime }(0),\hfill & \text{if}x=0.\hfill \end{array}\right),\text{where}p\ne q,\end{array}$$ we investigate the following property: \(\begin{aligned}(D_{p, q}^{n}f)(0)= & \lim _{x\rightarrow 0}D_{p, q}^{n}f(x)=\frac{f^{(n)}(0)((p, q); (p, q))_{n}}{n!(p-q)^n}\\= & \frac{f^{(n)}(0)[n]_{p, q}!}{n!}, \quad n\in \{1, 2, \ldots \},\end{aligned}\) $$\begin{array}{cc}\hfill (D_{p,q}^{n}f)(0)=& \underset{x\to 0}{lim}{D}_{p,q}^{n}f(x)=\frac{f^{(n)}(0){((p,q);(p,q))}_n}{n!{(p-q)}^n}\hfill \\ \hfill =& \frac{f^{(n)}(0){[n]}_{p,q}!}{n!},n\in \{1,2,\dots \},\hfill \end{array}$$ for which \(f^{(n)}(0)\) $f^{(n)}(0)$ exists. The real variable and the complex variable cases are considered. Specializing to the case \(p=1\) $p=1$ , we recover the property for the q-derivative operator as obtained by J. Koekoek and R. Koekoek in 1993, in related paper. Furthermore, some applications were discussed.