  We introduce a finite difference and $q$-difference analogues of the
Asymptotic Iteration Method of Ciftci, Hall, and Saad. We give necessary, and
sufficient condition for the existence of a polynomial solution to a general
linear second-order difference or $q$-difference equation subject to a
"terminating condition", which is precisely defined. When a difference or
$q$-difference equation has a polynomial solution, we show how to find the
second solution.
