q-difference equation

Summary

A $q$-difference equation relates the values of an unknown function at the points $x, qx, q^2 x, \dots$, where $q \ne 0, 1$ is a fixed number. It is a difference equation in which the shift $x \mapsto x + 1$ is replaced by the dilation $x \mapsto qx$. Equivalently, it can be written with the q-derivative in place of the ordinary derivative. Such equations govern $q$-analogs of special functions, such as basic hypergeometric series, and they tend formally to differential equations as $q \to 1$.