q-exponential

Summary

The $q$-exponential $e_q(z) = \sum_{n \ge 0} z^n / [n]_q!$ is the q-analog of the exponential function. It is the solution of $D_q f = f$, $f(0) = 1$, for the q-derivative $D_q$, just as $e^z$ solves $f' = f$. By the q-binomial theorem it is an infinite product, and it tends to $e^z$ as $q \to 1$. A second $q$-exponential $E_q$ satisfies $e_q(z) E_q(-z) = 1$, which replaces the rule $e^z e^{-z} = 1$.