Q-derivative

The $q$-derivative is $$\left(\dfrac{d}{dx} \right)_q f(x) =D_q\{f\}(x)=\left\{ \begin{array}{ll} \dfrac{f(qx)-f(x)}{qx-x} &; z \neq 0 \\ f'(0) &; z=0. \end{array} \right.$$

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.

References

• D.S. McAnally: q-exponential and q-gamma functions. I. q-exponential functions (1994)... (previous)... (next) $(2.2)$