Q-derivative
From specialfunctionswiki
The $q$-derivative is $$\left(\dfrac{d}{dx} \right)_q f(x) =D_q\{f\}(x)=\left\{ \begin{array}{ll} \dfrac{f(qx)-f(x)}{qx-x} &; x \neq 0 \\ f'(0) &; x=0. \end{array} \right.$$
Contents
Properties
Theorem
The following formula holds: $$D_q(z^n)=[n]_q z^{n-1},$$ where $D_q$ denotes the $q$-derivative and $[n]_q$ denotes the $q$-number.
