Q-derivative

The $q$-derivative is defined by $$\dfrac{\mathrm{d}_qf}{\mathrm{d}_qz}=\left\{ \begin{array}{ll} \dfrac{f(qz)-f(z)}{(q-1)z}, & \quad z \neq 0 \\ f'(0), & \quad z=0, \end{array} \right.$$ where $f'(0)$ denotes the derivative.

Properties

Relationship between q-derivative and derivative
q-derivative power rule

References

• D.S. McAnally: q-exponential and q-gamma functions. I. q-exponential functions (1994)... (previous)... (next) $(2.1)$
• 2002: Victor Kac and Pokman Cheung: Quantum Calculus ... (previous) ... (next) $(1.5)$