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

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Abstract In this note, for the ( p , q )-derivative operator $$D_{p, q}$$ D p , q defined by $$\begin{aligned}D_{p, q}f(x)={\left\{ \begin{array}{ll} \frac{f(px)-f(qx)}{(p-q)x} & \text {if } x\ne 0;\\ f'(0), & \text {if}~x=0. \end{array}\right. }, \quad \text { where } p\ne q,\end{aligned}$$ D p , q f ( x ) = f ( p x ) - f ( q x ) ( p - q ) x if x ≠ 0 ; f ′ ( 0 ) , if x = 0 . , where p ≠ q , we investigate the following property: $$\begin{aligned}(D_{p, q}^{n}f)(0)= & \lim _{x\rightarrow 0}D_{p, q}^{n}f(x)=\frac{f^{(n)}(0)((p, q); (p, q))_{n}}{n!(p-q)^n}\\= & \frac{f^{(n)}(0)[n]_{p, q}!}{n!}, \quad n\in \{1, 2, \ldots \},\end{aligned}$$ ( D p , q n f ) ( 0 ) = lim x → 0 D p , q n f ( x ) = f ( n ) ( 0 ) ( ( p , q ) ; ( p , q ) ) n n ! ( p - q ) n = f ( n ) ( 0 ) [ n ] p , q ! n ! , n ∈ { 1 , 2 , … } , 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.