Quantum calculus

Summary

Quantum calculus is calculus without limits. It replaces the derivative by a difference quotient taken at a fixed scale. The q-derivative compares $f(qx)$ with $f(x)$, and the $h$-derivative compares $f(x + h)$ with $f(x)$. The matching antiderivatives are the Jackson integral and finite sums. It is the natural language for q-analogs in combinatorics, for basic hypergeometric series and for q-difference equations. It recovers ordinary calculus in the limit $q \to 1$ or $h \to 0$.