# What is Quantum Calculus

Consider the following expression:

$$\frac{f\left(x\right)-f\left({x}_{0}\right)}{x-{x}_{0}}$$As $x$ approaches ${x}_{0}$, the limit, if it exists, gives the familiar definition of the derivative $\frac{df}{dx}$ of a function $f\left(x\right)$ at $x={x}_{0}$. However, if we take $x=\mathrm{q}{x}_{0}$. or $x={x}_{0}+\mathrm{h}$. where $\mathrm{q}$ is a fixed number different than $1$, and $\mathrm{h}$ a fixed number different from $0$, and do not take the limit, we enter the fascinating world of quantum calculus: The corresponding expressions are the definitions of

q-derivative andh-derivative of $f\left(x\right)$. Beginning with these two definitions [...] two types of quantum calculus, theq-calculus andh-calculus [are developed].

-- Victor G. Kac , Pokman Cheung

from "Quantum Calculus"

Quoted on Tue Nov 20th, 2012