A discrete and q asymptotic iteration method

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.